Solve {l}{(4-y)^2}{-y(y+1)^2}

$$\left. \begin{array} { l } { ( 4 - y ) ^ { 2 } } \\ { - y ( y + 1 ) ^ { 2 } } \end{array} \right.$$

Evaluate

$$-y\left(y+1\right)^{2}+\left(4-y\right)^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-y\right)^{2}.

$$16-8y+y^{2}-y\left(y+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+1\right)^{2}.

$$16-8y+y^{2}-y\left(y^{2}+2y+1\right)$$

Use the distributive property to multiply y by y^{2}+2y+1.

$$16-8y+y^{2}-\left(y^{3}+2y^{2}+y\right)$$

To find the opposite of y^{3}+2y^{2}+y, find the opposite of each term.

$$16-8y+y^{2}-y^{3}-2y^{2}-y$$

Combine y^{2} and -2y^{2} to get -y^{2}.

$$16-8y-y^{2}-y^{3}-y$$

Combine -8y and -y to get -9y.

$$16-9y-y^{2}-y^{3}$$

Expand

$$16-9y-y^{2}-y^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-y\right)^{2}.

$$16-8y+y^{2}-y\left(y+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+1\right)^{2}.

$$16-8y+y^{2}-y\left(y^{2}+2y+1\right)$$

Use the distributive property to multiply y by y^{2}+2y+1.

$$16-8y+y^{2}-\left(y^{3}+2y^{2}+y\right)$$

To find the opposite of y^{3}+2y^{2}+y, find the opposite of each term.

$$16-8y+y^{2}-y^{3}-2y^{2}-y$$

Combine y^{2} and -2y^{2} to get -y^{2}.

$$16-8y-y^{2}-y^{3}-y$$

Combine -8y and -y to get -9y.

$$16-9y-y^{2}-y^{3}$$