Use the formula for solving the square of a difference, \(\displaystyle \small (x-y)^{2}=x^{2}-2xy+y^{2}\) . In this case, \(\displaystyle \small \small \small (3a-2b)^{2}=9a^{2}-12ab+4b^{2}\)

To multiply a difference squared, square the first term and add two times the multiplication of the two terms. Then add the second term squared.

\(\displaystyle (2x^)^2+2(2x)(-6)+(-6)^2=4x^2-24x+36\)

First, we need to factor the numerator and denominator separately and cancel out similar terms. We will start with the numerator because it can be factored easily as the difference of two squares. 

\(\displaystyle x^2 - 25 = (x + 5)(x - 5)\)  

Now factor the quadratic in the denominator.

\(\displaystyle x^2 + 11x + 30 = (x + 5)(x + 6)\)

Substitute these factorizations back into the original expression.

\(\displaystyle \frac{(x+5)(x-5)}{(x+5)(x+6)}\)

The \(\displaystyle (x+5)\) terms cancel out, leaving us with the following answer:

\(\displaystyle \frac{x-5}{x+6}\)

2 raised to the power of 5 is the same as multiplying 2 by itself 5 times so:

2⁵ = 2x2x2x2x2 = 32

Then, 5x2 must first be multiplied before taking the exponent, yielding 10² = 100.

100 + 32 = 132

To multiply a difference squared, square the first term and add two times the multiplication of the two terms. Then add the second term squared.

\(\displaystyle (3x)^2+2(3x)(5)+(5)^2=9x^2+30x+25\)

Use the square of a sum pattern, substituting \(\displaystyle 4x\) for \(\displaystyle A\) and \(\displaystyle 7y\) for \(\displaystyle B\) in the pattern:

\(\displaystyle (A+ B)^{2}= A^{2}+ 2AB + B^{2}\)

\(\displaystyle (4x + 7y)^{2}= (4x)^{2}+ 2 (4x) (7y) + (7y)^{2}\)

\(\displaystyle = 16x^{2}+ 56xy + 49y^{2}\)

Use the square of a sum pattern, substituting \(\displaystyle 5 \sqrt{x}\) for \(\displaystyle A\) and \(\displaystyle 3 \sqrt{y}\) for \(\displaystyle B\) in the pattern:

\(\displaystyle (A+ B)^{2}= A^{2}+ 2AB + B^{2}\)

\(\displaystyle (5 \sqrt{x}+3 \sqrt{y})^{2}= (5 \sqrt{x})^{2}+ 2 (5 \sqrt{x})\left (3 \sqrt{y} \right ) + \left (3 \sqrt{y} \right )^{2}\)

\(\displaystyle (5 \sqrt{x}+3 \sqrt{y})^{2}= 25x+ 30 \sqrt{xy} + 9y\)

or 

\(\displaystyle 25x + 9y+ 30 \sqrt{xy}\)

Multiply vertically as follows:

                    \(\displaystyle \begin{matrix} x^{2}+ \; \; \; x +\; \; \; 7\\\underline{x^{2}+\; \; \; x +\; \; \; 7} \end{matrix}\)

                    \(\displaystyle 7 x^{2}+7 x +49\)

          \(\displaystyle x^{3}+ x^{2} + 7x\)

\(\displaystyle \underline{x^{4}+ x^{3}+7 x^{2} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }\)

\(\displaystyle x^{4}+2x^{3}+15x^{2}+14x+49\)

Use the square of a sum pattern, substituting \(\displaystyle y\) for \(\displaystyle A\) and \(\displaystyle \sqrt{17}\) for \(\displaystyle B\) in the pattern:

\(\displaystyle (A+ B)^{2}= A^{2}+ 2AB + B^{2}\)

\(\displaystyle (y + \sqrt{17})^{2}= y^{2}+ 2y \sqrt{17} + (\sqrt{17})^{2}\)

\(\displaystyle (y + \sqrt{17})^{2}= y^{2}+ 2y \sqrt{17} + 17\)

This is not equivalent to any of the given choices.

Use the square of a sum pattern, substituting \(\displaystyle 5 \sqrt{x}\) for \(\displaystyle A\) and \(\displaystyle 3 \sqrt{y}\) for \(\displaystyle B\) in the pattern:

\(\displaystyle (A+ B)^{2}= A^{2}+ 2AB + B^{2}\)

\(\displaystyle \left (\frac{1}{4}x + \frac{1}{7} \right )^{2}= \left (\frac{1}{4}x \right )^{2}+ 2 \left (\frac{1}{4}x \right )\left (\frac{1}{7} \right ) + \left (\frac{1}{7} \right )^{2}\)

\(\displaystyle = \frac{1}{16}x ^{2}+ \frac{1}{14}x +\frac{1}{49}\)