When subtracting polynomials, it's helpful to remember that the "minus sign" gets distributed. It's as if the two polynomials are being added and a -1 is in front of the second polynomial.

\(\displaystyle (4x^{2}-6x^{5}-2){\color{Blue} +}{\color{Red} (-1)}(x^{5}+3x^{3}-4x^{2}-2)\)

This -1 will get multiplied to all the terms in the second polynomial that is being subtracted from the first, so it becomes \(\displaystyle ({\color{red} -}x^{5}{\color{Red} -}3x^{3}{\color{Red} +}4x^{2}{\color{Red} +}2)\). You may note that by multiplying by -1, every term in the polynomial switches its original sign. The problem then becomes:

\(\displaystyle (4x^{2}-6x^{5}-2)+(-x^{5}-3x^{3}+4x^{2}+2)\)

From here, in order to simplify, because there's no equal sign, it can be deduced that we are not working toward a solution for x. The original problem is presented as an expression so an expression as an answer will be expected. In order to work towards a final simplified expression, like terms must be collected. This will provide the final answer.

\(\displaystyle {\color{Magenta} 4x^{2}}-{\color{DarkRed} 6x^{5}}{\color{Teal} -2}+{\color{DarkRed} -x^{5}}-3x^{3}+{\color{Magenta} 4x^{2}}{\color{Teal} +2}\)

\(\displaystyle {\color{DarkRed} -7x^{5}}-3x^{3}+{\color{Magenta} 8x^{2}}\) 

Begin by distributing the subtraction of the second term in this question:

\(\displaystyle 3x^2 + 5x +10 -10x^2+15x+3\)

Now, you merely need to combine like terms:

\(\displaystyle 3x^2 -10x^2 + 5x +15x +10 +3\)

\(\displaystyle -7x^2 + 20x +13\)

To solve this equation, we substitute \(\displaystyle (x+a)\) in for every instance of \(\displaystyle x\) seen in the original equation \(\displaystyle f(x)=2x^{2}-3\).

Therefore the new equation would read 

\(\displaystyle f(x)=2x^{2}-3\rightarrow f(x+a) = 2(x + a)^{2} -3\)

Now we must square the expression \(\displaystyle x+a\). To do this, you must multiply the expression by itself. Therefore:

\(\displaystyle (x+a)^{2}=(x+a)\cdot(x+a)\)

\(\displaystyle (x+a)\cdot(x+a)=(x\cdot x)+(x\cdot a)+(x\cdot a)+(a\cdot a)\)

\(\displaystyle (x\cdot x)+(x\cdot a)+(x\cdot a)+(a\cdot a)=x^{2}+2ax+a^{2}\)

We must now plug in our new value for \(\displaystyle (x+a)^{2}\) into our original equation in place of \(\displaystyle x^{2}\).

\(\displaystyle f(x)=2(x^{2}+2ax+a^{2})-3\)

Now we must distribute the \(\displaystyle 2\) into \(\displaystyle (x^{2}+2ax+a^{2})\). To do this, you multiply each expression within the parenthesis by \(\displaystyle 2\):

\(\displaystyle 2(x^{2}+2ax+a^{2})=2x^{2}+4ax+2a^{2}\)

Therefore, our answer is \(\displaystyle f(x)=2x^{2}+4ax+2a^{2}-3\).

To answer this question, we must distribute the \(\displaystyle a\) to the rest of the variables \(\displaystyle b\), \(\displaystyle c\), and \(\displaystyle d\) that are within the brackets.

To distribute a variable or number, you multiply that value with every other value within the brackets or parentheses. So, for this data:

\(\displaystyle a[(b-c)+d] = [[(a\cdot b)-(a\cdot c)] + (a\cdot d)]\)

We then simplify the expression by combining the variables we are multiplying together into expressions. For this data:

\(\displaystyle [[(a\cdot b)-(a\cdot c)] + (a\cdot d)] = ab-ac+ad\)

Be sure to keep all of your operations the same within the problem itself, unless the number being distributed is negative, which will then switch the signs with the brackets from positive to negative or negative to positive. 

Therefore, our answer is \(\displaystyle ab-ac+ad\).

To answer this question, we are solving for the values of \(\displaystyle x\) that make this equation true.

To this, we need to get \(\displaystyle x\) on a side by itself so we can evaluate it. To do this, we first add \(\displaystyle 25x\) to both sides so that we can then begin to deal with the \(\displaystyle x^{2}\) value. So, for this data:

\(\displaystyle x^{2}-25x=0\rightarrow x^{2}=25x\)

\(\displaystyle x^{2}\)can also be written as \(\displaystyle x\cdot x\). Therefore:

\(\displaystyle x^{2}=25x\rightarrow x\cdot x=25x\)

Now we can divide both sides by \(\displaystyle x\) and find the value of \(\displaystyle x\).

\(\displaystyle \frac{x\cdot x}{x}=\frac{25x}{x}\rightarrow x=25\)

Therefore, the answer to this question is \(\displaystyle x=25\)