Solve left(x-2right)left(x+3right)^2+2left(x-2right)^2left(x+3right)

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

Evaluate

$$\left(x-2\right)\left(3x-1\right)\left(x+3\right)$$

Solution Steps

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

$$\left(x-2\right)\left(x^{2}+6x+9\right)+2\left(x-2\right)^{2}\left(x+3\right)$$

Use the distributive property to multiply x-2 by x^{2}+6x+9 and combine like terms.

$$x^{3}+4x^{2}-3x-18+2\left(x-2\right)^{2}\left(x+3\right)$$

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

$$x^{3}+4x^{2}-3x-18+2\left(x^{2}-4x+4\right)\left(x+3\right)$$

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

$$x^{3}+4x^{2}-3x-18+\left(2x^{2}-8x+8\right)\left(x+3\right)$$

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

$$x^{3}+4x^{2}-3x-18+2x^{3}-2x^{2}-16x+24$$

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

$$3x^{3}+4x^{2}-3x-18-2x^{2}-16x+24$$

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

$$3x^{3}+2x^{2}-3x-18-16x+24$$

Combine -3x and -16x to get -19x.

$$3x^{3}+2x^{2}-19x-18+24$$

Add -18 and 24 to get 6.

$$3x^{3}+2x^{2}-19x+6$$

Expand

$$3x^{3}+2x^{2}-19x+6$$

Solution Steps

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

$$\left(x-2\right)\left(x^{2}+6x+9\right)+2\left(x-2\right)^{2}\left(x+3\right)$$

Use the distributive property to multiply x-2 by x^{2}+6x+9 and combine like terms.

$$x^{3}+4x^{2}-3x-18+2\left(x-2\right)^{2}\left(x+3\right)$$

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

$$x^{3}+4x^{2}-3x-18+2\left(x^{2}-4x+4\right)\left(x+3\right)$$

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

$$x^{3}+4x^{2}-3x-18+\left(2x^{2}-8x+8\right)\left(x+3\right)$$

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

$$x^{3}+4x^{2}-3x-18+2x^{3}-2x^{2}-16x+24$$

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

$$3x^{3}+4x^{2}-3x-18-2x^{2}-16x+24$$

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

$$3x^{3}+2x^{2}-3x-18-16x+24$$

Combine -3x and -16x to get -19x.

$$3x^{3}+2x^{2}-19x-18+24$$

Add -18 and 24 to get 6.

$$3x^{3}+2x^{2}-19x+6$$