Distribute each fraction: $$\frac{1}{2}(6x - 8) + \frac{3}{4}(4x + 12) = 3x - 4 + 3x + 9 = 6x + 5$$. Choice B results from calculating $$-4 + 9 = 4$$ instead of $$5$$. Choice C comes from incorrectly adding $$3x + 3x = 5x$$. Choice D results from errors in both distribution and combining terms.

Subtracting $$2(3x - 5)$$ from $$4(2x + 3)$$ means: $$4(2x + 3) - 2(3x - 5) = 8x + 12 - 6x + 10 = 2x + 22$$. The coefficient of $$x$$ is $$2$$. Choice B comes from adding instead of subtracting: $$8x + 6x = 14x$$. Choice C results from calculating $$6x - 8x = -2x$$. Choice D comes from incorrectly distributing.

To verify factoring is correct, use the distributive property to expand $$6(2x + 3y - 1) = 12x + 18y - 6$$, confirming it matches the original expression. Choice A (commutative) deals with order, choice B (associative) deals with grouping, and choice D (identity) is about value preservation but doesn't describe the verification method.

First combine like terms: $$15xy - 25x + 10x = 15xy - 15x$$. Factor out the GCF of $$5x$$: $$15xy - 15x = 5x(3y - 3)$$. Choice A fails to combine $$-25x + 10x$$ before factoring. Choice C uses a non-integer factor when $$5x$$ is available. Choice D factors out only $$x$$ instead of the complete GCF $$5x$$.

When you encounter factoring problems, remember that factoring means rewriting an expression as a product of simpler terms. The key is understanding when factoring is correct versus when it's complete.

The expression $$ax + bx + c$$ can indeed be factored as $$x(a + b) + c$$ because you're pulling out the common factor $$x$$ from the first two terms. This is mathematically valid regardless of the values of $$a$$, $$b$$, or $$c$$. However, this doesn't mean you've found the greatest common factor of all terms.

Let's examine why the other options miss the mark. Choice A incorrectly claims this only works when $$c = 0$$. While having $$c = 0$$ would allow complete factoring as $$x(a + b)$$, the partial factoring $$x(a + b) + c$$ is still mathematically correct when $$c \neq 0$$. Choice B gets the relationship backwards—when $$c = 0$$, you can factor completely, making the factoring more complete, not less. Choice C focuses on the coefficient $$a + b = 1$$, but this restriction isn't necessary for the factoring to be valid.

Choice D correctly identifies that this factoring is always mathematically sound but acknowledges it may not represent the most simplified form. For instance, if all terms share a common factor, you haven't fully factored until you extract that greatest common factor.

Study tip: When factoring, always check if you can pull out more common factors after your first step. Complete factoring means finding the greatest common factor of all terms, not just some of them.

To verify any factorization, expand it using the distributive property: $$7(3a - 2b + 5c) = 21a - 14b + 35c$$, which matches the original expression. While choices A, B, and D describe properties of correct factorization, choice C describes the direct verification method that confirms the factorization is mathematically equivalent to the original expression.

When you encounter expressions that need to be added together, the key is to expand each expression first using the distributive property, then combine like terms systematically.

Let's expand the first expression: $$3(x - 2) + 5x$$. Using the distributive property, $$3(x - 2) = 3x - 6$$, so this becomes $$3x - 6 + 5x = 8x - 6$$.

For the second expression: $$2(3 - x) - 4$$. Distributing gives us $$2(3 - x) = 6 - 2x$$, so this becomes $$6 - 2x - 4 = 2 - 2x$$.

Now we add the expanded expressions: $$(8x - 6) + (2 - 2x)$$. Combining like terms: $$8x - 2x - 6 + 2 = 6x - 4$$.

Looking at the wrong answers: Choice A ($$6x - 6$$) likely comes from incorrectly combining the constant terms as $$-6 - 4 = -10$$ instead of $$-6 + 2 = -4$$. Choice B ($$6x + 2$$) results from sign errors when distributing or combining terms. Choice C ($$8x - 4$$) happens when you correctly handle the constants but forget to combine the $$x$$ terms: $$8x - 2x = 6x$$, not $$8x$$.

The correct answer is D: $$6x - 4$$.

Strategy tip: Always expand completely before combining, and double-check your signs when distributing negative terms. Work step-by-step rather than trying to do multiple operations mentally—this prevents the sign errors that create most wrong answers in these problems.

This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, (2x - 3/4) + (-5x + 1/2) combines x terms (2 - 5 = -3x) and constants (-3/4 + 2/4 = -1/4), giving -3x - 1/4. In this case, the correct simplification is -3x - 1/4. A common error is sign error in adding fractions without a common denominator, such as -3/4 + 1/2 = -3/4 + 1/2 = 1/4 instead of -1/4. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.

This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). For example, (3x+5)+(2x-3) combines like terms: 3x+2x=5x, 5+(-3)=2, giving 5x+2; or (4x+7)-(2x+3) distributes the negative: 4x+7-2x-3, combines to 2x+4; or factor 8x+12: GCF=4, so 4(2x+3). Factor completely by finding GCF 6: 12x/6 = 2x, 18/6 = 3, giving 6(2x + 3), which is choice A. A common error is factoring partially, like pulling out only 3 to get 3(4x + 6) without checking for a larger GCF. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.

This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). A specific example is (4x+7)-(2x+3): distribute negative 4x+7-2x-3, combine 2x+4. For this problem, distribute negative: 1.2x - 4.8 - 0.7x - 1.5, combine x terms: 1.2x - 0.7x = 0.5x, constants: -4.8 - 1.5 = -6.3, giving 0.5x - 6.3. A common error is sign error in subtraction, like not distributing the negative to the constant term. Strategy: for adding/subtracting, distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); common mistakes include distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3).