Solve 2left(x+2right)^3-4left(x+2right)^2+7(x+2)-3

$$2 { \left(x+2 \right) }^{ 3 } -4 { \left(x+2 \right) }^{ 2 } +7(x+2)-3$$

Evaluate

$$2x^{3}+8x^{2}+15x+11$$

Solution Steps

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

$$2\left(x^{3}+6x^{2}+12x+8\right)-4\left(x+2\right)^{2}+7\left(x+2\right)-3$$

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

$$2x^{3}+12x^{2}+24x+16-4\left(x+2\right)^{2}+7\left(x+2\right)-3$$

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

$$2x^{3}+12x^{2}+24x+16-4\left(x^{2}+4x+4\right)+7\left(x+2\right)-3$$

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

$$2x^{3}+12x^{2}+24x+16-4x^{2}-16x-16+7\left(x+2\right)-3$$

Combine 12x^{2} and -4x^{2} to get 8x^{2}.

$$2x^{3}+8x^{2}+24x+16-16x-16+7\left(x+2\right)-3$$

Combine 24x and -16x to get 8x.

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

Subtract 16 from 16 to get 0.

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

Use the distributive property to multiply 7 by x+2.

$$2x^{3}+8x^{2}+8x+7x+14-3$$

Combine 8x and 7x to get 15x.

$$2x^{3}+8x^{2}+15x+14-3$$

Subtract 3 from 14 to get 11.

$$2x^{3}+8x^{2}+15x+11$$

Expand

$$2x^{3}+8x^{2}+15x+11$$

Solution Steps

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

$$2\left(x^{3}+6x^{2}+12x+8\right)-4\left(x+2\right)^{2}+7\left(x+2\right)-3$$

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

$$2x^{3}+12x^{2}+24x+16-4\left(x+2\right)^{2}+7\left(x+2\right)-3$$

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

$$2x^{3}+12x^{2}+24x+16-4\left(x^{2}+4x+4\right)+7\left(x+2\right)-3$$

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

$$2x^{3}+12x^{2}+24x+16-4x^{2}-16x-16+7\left(x+2\right)-3$$

Combine 12x^{2} and -4x^{2} to get 8x^{2}.

$$2x^{3}+8x^{2}+24x+16-16x-16+7\left(x+2\right)-3$$

Combine 24x and -16x to get 8x.

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

Subtract 16 from 16 to get 0.

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

Use the distributive property to multiply 7 by x+2.

$$2x^{3}+8x^{2}+8x+7x+14-3$$

Combine 8x and 7x to get 15x.

$$2x^{3}+8x^{2}+15x+14-3$$

Subtract 3 from 14 to get 11.

$$2x^{3}+8x^{2}+15x+11$$