Solve x^{2}-(x^{2}+4)(2-x)(2+x)-(2+x+x^2)^2-2(2-x)^3+16(x^2+2)

$$x ^ { 2 } - ( x ^ { 2 } + 4 ) ( 2 - x ) ( 2 + x ) - ( 2 + x + x ^ { 2 } ) ^ { 2 } - 2 ( 2 - x ) ^ { 3 } + 16 ( x ^ { 2 } + 2 )$$

Evaluate

$$20x-4$$

Solution Steps

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

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

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

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

To find the opposite of -x^{4}+16, find the opposite of each term.

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

Square 2+x+x^{2}.

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

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

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

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

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

Combine x^{2} and -5x^{2} to get -4x^{2}.

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

Subtract 4 from -16 to get -20.

$$-4x^{2}-20-2x^{3}-4x-2\left(2-x\right)^{3}+16\left(x^{2}+2\right)$$

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

$$-4x^{2}-20-2x^{3}-4x-2\left(8-12x+6x^{2}-x^{3}\right)+16\left(x^{2}+2\right)$$

Use the distributive property to multiply -2 by 8-12x+6x^{2}-x^{3}.

$$-4x^{2}-20-2x^{3}-4x-16+24x-12x^{2}+2x^{3}+16\left(x^{2}+2\right)$$

Subtract 16 from -20 to get -36.

$$-4x^{2}-36-2x^{3}-4x+24x-12x^{2}+2x^{3}+16\left(x^{2}+2\right)$$

Combine -4x and 24x to get 20x.

$$-4x^{2}-36-2x^{3}+20x-12x^{2}+2x^{3}+16\left(x^{2}+2\right)$$

Combine -4x^{2} and -12x^{2} to get -16x^{2}.

$$-16x^{2}-36-2x^{3}+20x+2x^{3}+16\left(x^{2}+2\right)$$

Combine -2x^{3} and 2x^{3} to get 0.

$$-16x^{2}-36+20x+16\left(x^{2}+2\right)$$

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

$$-16x^{2}-36+20x+16x^{2}+32$$

Combine -16x^{2} and 16x^{2} to get 0.

$$-36+20x+32$$

Add -36 and 32 to get -4.

$$-4+20x$$

Expand

$$20x-4$$

Solution Steps

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

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

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

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

To find the opposite of -x^{4}+16, find the opposite of each term.

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

Square 2+x+x^{2}.

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

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

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

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

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

Combine x^{2} and -5x^{2} to get -4x^{2}.

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

Subtract 4 from -16 to get -20.

$$-4x^{2}-20-2x^{3}-4x-2\left(2-x\right)^{3}+16\left(x^{2}+2\right)$$

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

$$-4x^{2}-20-2x^{3}-4x-2\left(8-12x+6x^{2}-x^{3}\right)+16\left(x^{2}+2\right)$$

Use the distributive property to multiply -2 by 8-12x+6x^{2}-x^{3}.

$$-4x^{2}-20-2x^{3}-4x-16+24x-12x^{2}+2x^{3}+16\left(x^{2}+2\right)$$

Subtract 16 from -20 to get -36.

$$-4x^{2}-36-2x^{3}-4x+24x-12x^{2}+2x^{3}+16\left(x^{2}+2\right)$$

Combine -4x and 24x to get 20x.

$$-4x^{2}-36-2x^{3}+20x-12x^{2}+2x^{3}+16\left(x^{2}+2\right)$$

Combine -4x^{2} and -12x^{2} to get -16x^{2}.

$$-16x^{2}-36-2x^{3}+20x+2x^{3}+16\left(x^{2}+2\right)$$

Combine -2x^{3} and 2x^{3} to get 0.

$$-16x^{2}-36+20x+16\left(x^{2}+2\right)$$

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

$$-16x^{2}-36+20x+16x^{2}+32$$

Combine -16x^{2} and 16x^{2} to get 0.

$$-36+20x+32$$

Add -36 and 32 to get -4.

$$-4+20x$$