Solve 2left(x-1right)^3+(x+2)left(x-1right)^2-22left(x+2right)^2(x-1)-24left(x+2right)^2

$$2 { \left(x-1 \right) }^{ 3 } +(x+2) { \left(x-1 \right) }^{ 2 } -22 { \left(x+2 \right) }^{ 2 } (x-1)-24 { \left(x+2 \right) }^{ 2 }$$

Evaluate

$$-19x^{3}-96x^{2}-93x-8$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$2\left(x^{3}-3x^{2}+3x-1\right)+\left(x+2\right)\left(x-1\right)^{2}-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Use the distributive property to multiply 2 by x^{3}-3x^{2}+3x-1.

$$2x^{3}-6x^{2}+6x-2+\left(x+2\right)\left(x-1\right)^{2}-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$2x^{3}-6x^{2}+6x-2+\left(x+2\right)\left(x^{2}-2x+1\right)-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Use the distributive property to multiply x+2 by x^{2}-2x+1 and combine like terms.

$$2x^{3}-6x^{2}+6x-2+x^{3}-3x+2-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

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

$$3x^{3}-6x^{2}+6x-2-3x+2-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Combine 6x and -3x to get 3x.

$$3x^{3}-6x^{2}+3x-2+2-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Add -2 and 2 to get 0.

$$3x^{3}-6x^{2}+3x-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

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

$$3x^{3}-6x^{2}+3x-22\left(x^{2}+4x+4\right)\left(x-1\right)-24\left(x+2\right)^{2}$$

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

$$3x^{3}-6x^{2}+3x-22\left(x^{2}+4x+4\right)\left(x-1\right)-24\left(x^{2}+4x+4\right)$$

Use the distributive property to multiply -24 by x^{2}+4x+4.

$$3x^{3}-6x^{2}+3x-22\left(x^{2}+4x+4\right)\left(x-1\right)-24x^{2}-96x-96$$

Use the distributive property to multiply -22 by x^{2}+4x+4.

$$3x^{3}-6x^{2}+3x+\left(-22x^{2}-88x-88\right)\left(x-1\right)-24x^{2}-96x-96$$

Use the distributive property to multiply -22x^{2}-88x-88 by x-1 and combine like terms.

$$3x^{3}-6x^{2}+3x-22x^{3}-66x^{2}+88-24x^{2}-96x-96$$

Combine 3x^{3} and -22x^{3} to get -19x^{3}.

$$-19x^{3}-6x^{2}+3x-66x^{2}+88-24x^{2}-96x-96$$

Combine -6x^{2} and -66x^{2} to get -72x^{2}.

$$-19x^{3}-72x^{2}+3x+88-24x^{2}-96x-96$$

Combine -72x^{2} and -24x^{2} to get -96x^{2}.

$$-19x^{3}-96x^{2}+3x+88-96x-96$$

Combine 3x and -96x to get -93x.

$$-19x^{3}-96x^{2}-93x+88-96$$

Subtract 96 from 88 to get -8.

$$-19x^{3}-96x^{2}-93x-8$$

Expand

$$-19x^{3}-96x^{2}-93x-8$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$2\left(x^{3}-3x^{2}+3x-1\right)+\left(x+2\right)\left(x-1\right)^{2}-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Use the distributive property to multiply 2 by x^{3}-3x^{2}+3x-1.

$$2x^{3}-6x^{2}+6x-2+\left(x+2\right)\left(x-1\right)^{2}-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$2x^{3}-6x^{2}+6x-2+\left(x+2\right)\left(x^{2}-2x+1\right)-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Use the distributive property to multiply x+2 by x^{2}-2x+1 and combine like terms.

$$2x^{3}-6x^{2}+6x-2+x^{3}-3x+2-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

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

$$3x^{3}-6x^{2}+6x-2-3x+2-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Combine 6x and -3x to get 3x.

$$3x^{3}-6x^{2}+3x-2+2-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

Add -2 and 2 to get 0.

$$3x^{3}-6x^{2}+3x-22\left(x+2\right)^{2}\left(x-1\right)-24\left(x+2\right)^{2}$$

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

$$3x^{3}-6x^{2}+3x-22\left(x^{2}+4x+4\right)\left(x-1\right)-24\left(x+2\right)^{2}$$

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

$$3x^{3}-6x^{2}+3x-22\left(x^{2}+4x+4\right)\left(x-1\right)-24\left(x^{2}+4x+4\right)$$

Use the distributive property to multiply -24 by x^{2}+4x+4.

$$3x^{3}-6x^{2}+3x-22\left(x^{2}+4x+4\right)\left(x-1\right)-24x^{2}-96x-96$$

Use the distributive property to multiply -22 by x^{2}+4x+4.

$$3x^{3}-6x^{2}+3x+\left(-22x^{2}-88x-88\right)\left(x-1\right)-24x^{2}-96x-96$$

Use the distributive property to multiply -22x^{2}-88x-88 by x-1 and combine like terms.

$$3x^{3}-6x^{2}+3x-22x^{3}-66x^{2}+88-24x^{2}-96x-96$$

Combine 3x^{3} and -22x^{3} to get -19x^{3}.

$$-19x^{3}-6x^{2}+3x-66x^{2}+88-24x^{2}-96x-96$$

Combine -6x^{2} and -66x^{2} to get -72x^{2}.

$$-19x^{3}-72x^{2}+3x+88-24x^{2}-96x-96$$

Combine -72x^{2} and -24x^{2} to get -96x^{2}.

$$-19x^{3}-96x^{2}+3x+88-96x-96$$

Combine 3x and -96x to get -93x.

$$-19x^{3}-96x^{2}-93x+88-96$$

Subtract 96 from 88 to get -8.

$$-19x^{3}-96x^{2}-93x-8$$