Solve (x-8)^2-(x-2)^3-8(9+x^2)+x^2(x-1)

$$( x - 8 ) ^ { 2 } - ( x - 2 ) ^ { 3 } - 8 ( 9 + x ^ { 2 } ) + x ^ { 2 } ( x - 1 )$$

Evaluate

$$-2x\left(x+14\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64-\left(x-2\right)^{3}-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

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

$$x^{2}-16x+64-\left(x^{3}-6x^{2}+12x-8\right)-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$x^{2}-16x+64-x^{3}+6x^{2}-12x+8-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

Combine x^{2} and 6x^{2} to get 7x^{2}.

$$7x^{2}-16x+64-x^{3}-12x+8-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

Combine -16x and -12x to get -28x.

$$7x^{2}-28x+64-x^{3}+8-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

Add 64 and 8 to get 72.

$$7x^{2}-28x+72-x^{3}-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

Use the distributive property to multiply -8 by 9+x^{2}.

$$7x^{2}-28x+72-x^{3}-72-8x^{2}+x^{2}\left(x-1\right)$$

Subtract 72 from 72 to get 0.

$$7x^{2}-28x-x^{3}-8x^{2}+x^{2}\left(x-1\right)$$

Combine 7x^{2} and -8x^{2} to get -x^{2}.

$$-x^{2}-28x-x^{3}+x^{2}\left(x-1\right)$$

Use the distributive property to multiply x^{2} by x-1.

$$-x^{2}-28x-x^{3}+x^{3}-x^{2}$$

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

$$-x^{2}-28x-x^{2}$$

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

$$-2x^{2}-28x$$

Expand

$$-2x^{2}-28x$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64-\left(x-2\right)^{3}-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

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

$$x^{2}-16x+64-\left(x^{3}-6x^{2}+12x-8\right)-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$x^{2}-16x+64-x^{3}+6x^{2}-12x+8-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

Combine x^{2} and 6x^{2} to get 7x^{2}.

$$7x^{2}-16x+64-x^{3}-12x+8-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

Combine -16x and -12x to get -28x.

$$7x^{2}-28x+64-x^{3}+8-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

Add 64 and 8 to get 72.

$$7x^{2}-28x+72-x^{3}-8\left(9+x^{2}\right)+x^{2}\left(x-1\right)$$

Use the distributive property to multiply -8 by 9+x^{2}.

$$7x^{2}-28x+72-x^{3}-72-8x^{2}+x^{2}\left(x-1\right)$$

Subtract 72 from 72 to get 0.

$$7x^{2}-28x-x^{3}-8x^{2}+x^{2}\left(x-1\right)$$

Combine 7x^{2} and -8x^{2} to get -x^{2}.

$$-x^{2}-28x-x^{3}+x^{2}\left(x-1\right)$$

Use the distributive property to multiply x^{2} by x-1.

$$-x^{2}-28x-x^{3}+x^{3}-x^{2}$$

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

$$-x^{2}-28x-x^{2}$$

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

$$-2x^{2}-28x$$