Let's check each option: A) $$6(2x + 3y) = 12x + 18y$$ ✓. B) $$3(4x + 6y) = 12x + 18y$$ ✓. C) $$2(6x + 8y) = 12x + 16y ≠ 12x + 18y$$ ✗. D) $$18y + 12x = 12x + 18y$$ ✓. Choice C is incorrect because $$2 × 9 = 18$$, not $$2 × 8 = 16$$.

When you encounter expressions with parentheses and variables, you need to use the distributive property and combine like terms to find equivalent expressions.

Let's simplify the original expression $$4(2m + 3n) - 2(m + n)$$ step by step. First, distribute the 4: $$4(2m + 3n) = 8m + 12n$$. Next, distribute the -2: $$-2(m + n) = -2m - 2n$$. Now combine: $$8m + 12n - 2m - 2n$$. Finally, combine like terms: $$8m - 2m = 6m$$ and $$12n - 2n = 10n$$, giving us $$6m + 10n$$.

Now let's check each answer choice. Choice A gives $$6m + 10n$$ (correct) and $$2(3m + 4n) + 4n = 6m + 8n + 4n = 6m + 12n$$ (incorrect). Choice B provides $$8m + 12n - 2m - 2n$$ (this is the unsimplified form, which equals our answer) and $$6m + 10n$$ (correct). However, the question asks for expressions equivalent to the original, and the unsimplified form isn't really a separate equivalent expression—it's just an intermediate step.

Choice C offers $$6m + 10n$$ (correct) and $$2(3m + 5n) = 6m + 10n$$ (also correct). Choice D gives $$8m + 6n$$ (incorrect—wrong coefficients) and $$2(4m + 3n) = 8m + 6n$$ (same incorrect expression).

Remember: always simplify expressions completely by distributing first, then combining like terms. Double-check by expanding any factored forms in the answer choices.

When you see an expression with parentheses like this, you need to apply the distributive property step by step before combining like terms. The distributive property means you multiply the number outside the parentheses by each term inside.

Let's work through $$5(x - 2) + 3(2x + 1) - 4x$$ systematically. First, distribute the 5: $$5 \cdot x = 5x$$ and $$5 \cdot(-2) = -10$$, giving us $$5x - 10$$. Next, distribute the 3: $$3 \cdot 2x = 6x$$ and $$3 \cdot 1 = 3$$, giving us $$6x + 3$$. The last term, $$-4x$$, stays as is since there are no parentheses. After distributing everything, you get $$5x - 10 + 6x + 3 - 4x$$.

Choice A incorrectly ignores the coefficients outside the parentheses entirely, treating $$5(x - 2)$$ as just $$x - 2$$. Choice B partially applies the distributive property but forgets to multiply the constants inside the parentheses—it should be $$5 \cdot(-2) = -10$$, not just $$-2$$. Choice C jumps ahead by combining some terms during the distribution step, but the question specifically asks what the expression looks like "after distributing but before combining like terms."

Choice D correctly shows all multiplication completed: $$5x - 10 + 6x + 3 - 4x$$. This is exactly what you should have after applying the distributive property completely but before the final step of combining like terms.

Remember: distribute first, then combine like terms. Don't skip steps or you'll make careless errors.

When combining like terms, you add the coefficients: $$2x + 3x + x = (2 + 3 + 1)x = 6x$$. The student incorrectly squared the result. Choice B incorrectly applies exponent rules. Choice C shows a misunderstanding of combining like terms. Choice D incorrectly combines only some terms.

When you encounter algebraic expressions that claim to be equivalent, you're testing whether they simplify to the same form and produce identical results when you substitute values. This is a fundamental skill for working with algebraic expressions.

Let's verify these expressions by substituting $$a = 2$$ and $$b = 1$$. For the first expression $$8a + 12b$$: $$8(2) + 12(1) = 16 + 12 = 28$$. For the second expression $$4(2a + 3b)$$: $$4(2(2) + 3(1)) = 4(4 + 3) = 4(7) = 28$$. Both expressions equal 28, confirming they are equivalent.

You can also verify this algebraically by distributing the 4 in the second expression: $$4(2a + 3b) = 4 \cdot 2a + 4 \cdot 3b = 8a + 12b$$, which matches the first expression exactly.

Looking at the wrong answers: Choice A gives 20, which might result from calculation errors like $$8(2) + 12(1) = 16 + 4 = 20$$ if you mistakenly calculated $$12 \times 1 = 4$$. Choice B gives 24, which could come from errors like $$8(2) + 8(1) = 16 + 8 = 24$$ if you confused the coefficient 12 with 8. Choice C gives 32, which might result from $$8(2) + 16(1) = 16 + 16 = 32$$ if you doubled the coefficient 12 incorrectly.

When verifying equivalent expressions through substitution, always double-check your arithmetic carefully. Small calculation errors can lead you to conclude that equivalent expressions are different, or vice versa.

The distributive property states that $$a(b + c) = ab + ac$$. In reverse, $$ab + ac = a(b + c)$$. So $$3y + 3y + 3y = (3 + 3 + 3)y = 9y$$. Choice A incorrectly factors and uses the distributive property. Choice C incorrectly applies the associative property. Choice D correctly identifies the final steps but incorrectly names the first property used.

The greatest common factor of 15, 25, and 10 is 5. Factoring out 5: $$15a + 25b - 10c = 5(3a + 5b - 2c)$$. Choice B incorrectly shows $$-c$$ instead of $$-2c$$. Choice C uses 15 as a factor, but 15 doesn't divide evenly into 25 or 10. Choice D uses 10 as a factor, but 10 doesn't divide evenly into 15 or 25.

Using the distributive property in reverse, we can factor out the greatest common factor of 3: $$6x + 9y + 12z = 3(2x + 3y + 4z)$$. Choice B gives $$6x + 9y + 4z$$, not $$6x + 9y + 12z$$. Choice C expands to $$6x + 6y + 6z + 3y + 6z = 6x + 9y + 12z$$, which is actually equivalent but not the simplest factored form. Choice D expands to $$6x + 9y + 3z + 9z = 6x + 9y + 12z$$, which is also equivalent but not the simplest factored form.

When you encounter a perimeter problem involving algebraic expressions, remember that perimeter means adding up all the sides of a shape. For a rectangle, the perimeter formula is $$2 \times \text{length} + 2 \times \text{width}$$.

Let's work through the given expression $$2(3x + 4) + 2(2)$$ step by step. First, distribute the 2 into the first parentheses: $$2(3x + 4) = 6x + 8$$. Next, simplify the second term: $$2(2) = 4$$. Now combine these results: $$6x + 8 + 4 = 6x + 12$$.

Looking at the answer choices, option B correctly shows this process: $$6x + 8 + 4$$ which simplifies to $$6x + 12$$ units.

Option A makes a distribution error, writing $$6x + 4 + 4$$ instead of $$6x + 8 + 4$$. They forgot to distribute the 2 to both terms inside the parentheses. Option C shows $$3x + 4 + 2 + 2$$, which represents adding the length and width once each instead of twice each—this ignores the perimeter formula entirely. Option D gives $$6x + 6$$, which appears to come from incorrectly calculating $$2(3x + 4)$$ as $$6x + 4$$ and then adding $$2(2) = 4$$ to get $$6x + 8$$, but somehow ending up with $$6x + 6$$.

Study tip: When distributing multiplication over addition, make sure to multiply the number outside the parentheses by every term inside. Don't rush—distribute carefully and then combine like terms at the end.

This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression using the distributive property to expand 3(2x + 5) is 6x + 15, by multiplying 3 by 2x and 3 by 5. A common error is incomplete distribution, like 3(2x + 5)=6x + 5 missing the 15, or arithmetic wrong like 3×5=8. Expanding: distribute multiplier to every term inside parentheses (a(b+c)=ab+ac, don't miss any terms). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.