Solve (4-x^{2})^2(x^3-2)+x^3(-x^4-2)+32

$$( 4 - x ^ { 2 } ) ^ { 2 } ( x ^ { 3 } - 2 ) + x ^ { 3 } ( - x ^ { 4 } - 2 ) + 32$$

Evaluate

$$2x^{2}\left(8+7x-x^{2}-4x^{3}\right)$$

Solution Steps

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

$$\left(16-8x^{2}+\left(x^{2}\right)^{2}\right)\left(x^{3}-2\right)+x^{3}\left(-x^{4}-2\right)+32$$

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

$$\left(16-8x^{2}+x^{4}\right)\left(x^{3}-2\right)+x^{3}\left(-x^{4}-2\right)+32$$

Use the distributive property to multiply 16-8x^{2}+x^{4} by x^{3}-2.

$$16x^{3}-32-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{3}\left(-x^{4}-2\right)+32$$

Use the distributive property to multiply x^{3} by -x^{4}-2.

$$16x^{3}-32-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{3}\left(-x^{4}\right)-2x^{3}+32$$

Combine 16x^{3} and -2x^{3} to get 14x^{3}.

$$14x^{3}-32-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{3}\left(-x^{4}\right)+32$$

Add -32 and 32 to get 0.

$$14x^{3}-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{3}\left(-x^{4}\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$14x^{3}-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{7}\left(-1\right)$$

Combine x^{7} and x^{7}\left(-1\right) to get 0.

$$14x^{3}-8x^{5}+16x^{2}-2x^{4}$$

Expand

$$16x^{2}+14x^{3}-2x^{4}-8x^{5}$$

Solution Steps

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

$$\left(16-8x^{2}+\left(x^{2}\right)^{2}\right)\left(x^{3}-2\right)+x^{3}\left(-x^{4}-2\right)+32$$

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

$$\left(16-8x^{2}+x^{4}\right)\left(x^{3}-2\right)+x^{3}\left(-x^{4}-2\right)+32$$

Use the distributive property to multiply 16-8x^{2}+x^{4} by x^{3}-2.

$$16x^{3}-32-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{3}\left(-x^{4}-2\right)+32$$

Use the distributive property to multiply x^{3} by -x^{4}-2.

$$16x^{3}-32-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{3}\left(-x^{4}\right)-2x^{3}+32$$

Combine 16x^{3} and -2x^{3} to get 14x^{3}.

$$14x^{3}-32-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{3}\left(-x^{4}\right)+32$$

Add -32 and 32 to get 0.

$$14x^{3}-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{3}\left(-x^{4}\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$14x^{3}-8x^{5}+16x^{2}+x^{7}-2x^{4}+x^{7}\left(-1\right)$$

Combine x^{7} and x^{7}\left(-1\right) to get 0.

$$14x^{3}-8x^{5}+16x^{2}-2x^{4}$$