The degree of a term of a polynomial with one variable is the exponent of that variable. The terms of a polynomial in standard form are written in descending order of degree. Therefore, we rearrange the terms by their exponent, from 5 down to 0, noting that we can rewrite the \(\displaystyle x\) and constant terms with exponents 1 and 0, respectively:

\(\displaystyle 9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}\)

\(\displaystyle = 9x^{4} - 3x ^{2} + 8x^{5} -2x ^{1} + 7x ^{0} - 4x^{3}\)

\(\displaystyle =8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x ^{1}+ 7x^{0}\)

\(\displaystyle =8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x+ 7\)

\(\displaystyle \frac{(x^{2}+y^{2})^{2} - (x^{2} - y^{2})^{2}}{x^{4}y^{4}}\)

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

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

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

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

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

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

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

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

\(\displaystyle \frac{(x-y)^2+4xy}{x+y}=\frac{(x^2-2xy+y^2)+4xy}{x+y}\)

\(\displaystyle =\frac{(x^2-2xy+4xy+y^2)}{x+y}\)

\(\displaystyle =\frac{(x^2+2xy+y^2)}{x+y}\)

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

\(\displaystyle \frac{(x+\frac{1}{x})^3+(x-\frac{1}{x})^3}{\frac{2}{x}}\)

\(\displaystyle =\frac{(x^3+3x^2\times \frac{1}{x}+3x\times \frac{1}{x^2}+\frac{1}{x^3})+(x^3-3x^2\times \frac{1}{x}+3x\times \frac{1}{x^2}-\frac{1}{x^3})}{\frac{2}{x}}\)

\(\displaystyle =(x^3+x^3)+(3x-3x)+(\frac{3}{x}+\frac{3}{x})+(\frac{1}{x^3}-\frac{1}{x^3})\)

\(\displaystyle =\frac{2x^3+\frac{6}{x}}{\frac{2}{x}}\)

\(\displaystyle =\frac{\frac{2x^4+6}{x}}{\frac{2}{x}}\)

\(\displaystyle =\frac{2x^4+6}{2}\)

\(\displaystyle =\frac{2(x^4+3)}{2}\)

\(\displaystyle =x^4+3\)

Substitute \(\displaystyle x ^{2} + 3x\) for \(\displaystyle x\) in the definition:

\(\displaystyle g(x) = 2x - 5\)

\(\displaystyle g(x ^{2} + 3x ) = 2 \left ( x ^{2} + 3x \right ) - 5\)

\(\displaystyle g(x ^{2} + 3x ) = 2 \left ( x ^{2}\right ) + 2 \left ( 3x \right ) - 5\)

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

\(\displaystyle x^3+4x^2-2x^3+3x+2x^2+4\)

\(\displaystyle =(x^3-2x^3)+(4x^2+2x^2)+3x+4\)

\(\displaystyle =-x^3+6x^2+3x+4\)

We can expand the first term using FOIL:

\(\displaystyle (x+y)^2=x^2+2xy+y^2\)

\(\displaystyle (x+y)^2+2x^2-2y^2=x^2+2xy+y^2+2x^2-2y^2\)

Reorder the expression to group like-terms together.

\(\displaystyle (x^2+2x^2)+2xy+(y^2-2y^2)\)

Simplify by combining like-terms.

\(\displaystyle 3x^2+2xy-y^2\)

Expand each term by using FOIL:

\(\displaystyle (2x-y)^2-(x-2y)^2 = [(2x)^2-(2) (2x)(y)+(y)^2]-[(x)^2+(2) (x)(2y)-(2y)^2]\)

\(\displaystyle =4x^2-4xy+y^2-x^2+4xy-4y^2\)

Rearrange to group like-terms together.

\(\displaystyle (4x^2-x^2)+(-4xy+4xy)+(y^2-4y^2)\)

Simplify by combining like-terms.

\(\displaystyle 3x^2-3y^2\)

Start by reordering the expression to group like-terms together.

\(\displaystyle 4x^3y-x^2y^2-3x^3y+2x^2y^2\)

\(\displaystyle (4x^3y-3x^3y)+(-x^2y^2+2x^2y^2)\)

Combine like-terms to simplify.

\(\displaystyle (4x^3y-3x^3y)+(-x^2y^2+2x^2y^2)\)

\(\displaystyle x^3y+x^2y^2\)