Solve (x^{2}+4)(2-x)(2+x)-(2+x+x^2)^2-2(2-x)^2+16(x^2+2)

$$( x ^ { 2 } + 4 ) ( 2 - x ) ( 2 + x ) - ( 2 + x + x ^ { 2 } ) ^ { 2 } - 2 ( 2 - x ) ^ { 2 } + 16 ( x ^ { 2 } + 2 )$$

Evaluate

$$36+4x+9x^{2}-2x^{3}-2x^{4}$$

Solution Steps

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

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

Use the distributive property to multiply 2x^{2}-x^{3}+8-4x by 2+x and combine like terms.

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

Square 2+x+x^{2}.

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

To find the opposite of x^{4}+2x^{3}+5x^{2}+4x+4, find the opposite of each term.

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

Combine -x^{4} and -x^{4} to get -2x^{4}.

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

Subtract 4 from 16 to get 12.

$$-2x^{4}+12-2x^{3}-5x^{2}-4x-2\left(2-x\right)^{2}+16\left(x^{2}+2\right)$$

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

$$-2x^{4}+12-2x^{3}-5x^{2}-4x-2\left(4-4x+x^{2}\right)+16\left(x^{2}+2\right)$$

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

$$-2x^{4}+12-2x^{3}-5x^{2}-4x-8+8x-2x^{2}+16\left(x^{2}+2\right)$$

Subtract 8 from 12 to get 4.

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

Combine -4x and 8x to get 4x.

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

Combine -5x^{2} and -2x^{2} to get -7x^{2}.

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

Use the distributive property to multiply 16 by x^{2}+2.

$$-2x^{4}+4-2x^{3}-7x^{2}+4x+16x^{2}+32$$

Combine -7x^{2} and 16x^{2} to get 9x^{2}.

$$-2x^{4}+4-2x^{3}+9x^{2}+4x+32$$

Add 4 and 32 to get 36.

$$-2x^{4}+36-2x^{3}+9x^{2}+4x$$

Expand

$$36+4x+9x^{2}-2x^{3}-2x^{4}$$

Solution Steps

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

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

Use the distributive property to multiply 2x^{2}-x^{3}+8-4x by 2+x and combine like terms.

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

Square 2+x+x^{2}.

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

To find the opposite of x^{4}+2x^{3}+5x^{2}+4x+4, find the opposite of each term.

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

Combine -x^{4} and -x^{4} to get -2x^{4}.

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

Subtract 4 from 16 to get 12.

$$-2x^{4}+12-2x^{3}-5x^{2}-4x-2\left(2-x\right)^{2}+16\left(x^{2}+2\right)$$

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

$$-2x^{4}+12-2x^{3}-5x^{2}-4x-2\left(4-4x+x^{2}\right)+16\left(x^{2}+2\right)$$

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

$$-2x^{4}+12-2x^{3}-5x^{2}-4x-8+8x-2x^{2}+16\left(x^{2}+2\right)$$

Subtract 8 from 12 to get 4.

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

Combine -4x and 8x to get 4x.

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

Combine -5x^{2} and -2x^{2} to get -7x^{2}.

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

Use the distributive property to multiply 16 by x^{2}+2.

$$-2x^{4}+4-2x^{3}-7x^{2}+4x+16x^{2}+32$$

Combine -7x^{2} and 16x^{2} to get 9x^{2}.

$$-2x^{4}+4-2x^{3}+9x^{2}+4x+32$$

Add 4 and 32 to get 36.

$$-2x^{4}+36-2x^{3}+9x^{2}+4x$$