Solve 3(x-1)(x+1)^2-6(x-1)^2(x+1)

$$3 ( x - 1 ) ( x + 1 ) ^ { 2 } - 6 ( x - 1 ) ^ { 2 } ( x + 1 )$$

Evaluate

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

Solution Steps

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

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

Use the distributive property to multiply 3 by x-1.

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

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

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

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

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

Use the distributive property to multiply -6 by x^{2}-2x+1.

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

Use the distributive property to multiply -6x^{2}+12x-6 by x+1 and combine like terms.

$$3x^{3}+3x^{2}-3x-3-6x^{3}+6x^{2}+6x-6$$

Combine 3x^{3} and -6x^{3} to get -3x^{3}.

$$-3x^{3}+3x^{2}-3x-3+6x^{2}+6x-6$$

Combine 3x^{2} and 6x^{2} to get 9x^{2}.

$$-3x^{3}+9x^{2}-3x-3+6x-6$$

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

$$-3x^{3}+9x^{2}+3x-3-6$$

Subtract 6 from -3 to get -9.

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

Expand

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

Solution Steps

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

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

Use the distributive property to multiply 3 by x-1.

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

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

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

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

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

Use the distributive property to multiply -6 by x^{2}-2x+1.

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

Use the distributive property to multiply -6x^{2}+12x-6 by x+1 and combine like terms.

$$3x^{3}+3x^{2}-3x-3-6x^{3}+6x^{2}+6x-6$$

Combine 3x^{3} and -6x^{3} to get -3x^{3}.

$$-3x^{3}+3x^{2}-3x-3+6x^{2}+6x-6$$

Combine 3x^{2} and 6x^{2} to get 9x^{2}.

$$-3x^{3}+9x^{2}-3x-3+6x-6$$

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

$$-3x^{3}+9x^{2}+3x-3-6$$

Subtract 6 from -3 to get -9.

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