Solve (y^{2}-x)^2+y^2(2x-y^2)-(-x-3)^2=

$$( y ^ { 2 } - x ) ^ { 2 } + y ^ { 2 } ( 2 x - y ^ { 2 } ) - ( - x - 3 ) ^ { 2 } =$$

Evaluate

$$-6x-9$$

Solution Steps

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$y^{4}-2y^{2}x+x^{2}+y^{2}\left(2x-y^{2}\right)-\left(-x-3\right)^{2}$$

Use the distributive property to multiply y^{2} by 2x-y^{2}.

$$y^{4}-2y^{2}x+x^{2}+2y^{2}x-y^{4}-\left(-x-3\right)^{2}$$

Combine -2y^{2}x and 2y^{2}x to get 0.

$$y^{4}+x^{2}-y^{4}-\left(-x-3\right)^{2}$$

Combine y^{4} and -y^{4} to get 0.

$$x^{2}-\left(-x-3\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-x-3\right)^{2}.

$$x^{2}-\left(\left(-x\right)^{2}-6\left(-x\right)+9\right)$$

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}-\left(x^{2}-6\left(-x\right)+9\right)$$

Multiply -6 and -1 to get 6.

$$x^{2}-\left(x^{2}+6x+9\right)$$

To find the opposite of x^{2}+6x+9, find the opposite of each term.

$$x^{2}-x^{2}-6x-9$$

Combine x^{2} and -x^{2} to get 0.

$$-6x-9$$

Expand

$$-6x-9$$

Solution Steps

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$y^{4}-2y^{2}x+x^{2}+y^{2}\left(2x-y^{2}\right)-\left(-x-3\right)^{2}$$

Use the distributive property to multiply y^{2} by 2x-y^{2}.

$$y^{4}-2y^{2}x+x^{2}+2y^{2}x-y^{4}-\left(-x-3\right)^{2}$$

Combine -2y^{2}x and 2y^{2}x to get 0.

$$y^{4}+x^{2}-y^{4}-\left(-x-3\right)^{2}$$

Combine y^{4} and -y^{4} to get 0.

$$x^{2}-\left(-x-3\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-x-3\right)^{2}.

$$x^{2}-\left(\left(-x\right)^{2}-6\left(-x\right)+9\right)$$

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}-\left(x^{2}-6\left(-x\right)+9\right)$$

Multiply -6 and -1 to get 6.

$$x^{2}-\left(x^{2}+6x+9\right)$$

To find the opposite of x^{2}+6x+9, find the opposite of each term.

$$x^{2}-x^{2}-6x-9$$

Combine x^{2} and -x^{2} to get 0.

$$-6x-9$$