Solve =2(x+1)^3+4(x+1)^2-3(x+1)+6

$$= 2 ( x + 1 ) ^ { 3 } + 4 ( x + 1 ) ^ { 2 } - 3 ( x + 1 ) + 6$$

Evaluate

$$2x^{3}+10x^{2}+11x+9$$

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}.

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

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

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

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

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

Use the distributive property to multiply 4 by x^{2}+2x+1.

$$2x^{3}+6x^{2}+6x+2+4x^{2}+8x+4-3\left(x+1\right)+6$$

Combine 6x^{2} and 4x^{2} to get 10x^{2}.

$$2x^{3}+10x^{2}+6x+2+8x+4-3\left(x+1\right)+6$$

Combine 6x and 8x to get 14x.

$$2x^{3}+10x^{2}+14x+2+4-3\left(x+1\right)+6$$

Add 2 and 4 to get 6.

$$2x^{3}+10x^{2}+14x+6-3\left(x+1\right)+6$$

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

$$2x^{3}+10x^{2}+14x+6-3x-3+6$$

Combine 14x and -3x to get 11x.

$$2x^{3}+10x^{2}+11x+6-3+6$$

Subtract 3 from 6 to get 3.

$$2x^{3}+10x^{2}+11x+3+6$$

Add 3 and 6 to get 9.

$$2x^{3}+10x^{2}+11x+9$$

Expand

$$2x^{3}+10x^{2}+11x+9$$

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}.

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

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

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

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

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

Use the distributive property to multiply 4 by x^{2}+2x+1.

$$2x^{3}+6x^{2}+6x+2+4x^{2}+8x+4-3\left(x+1\right)+6$$

Combine 6x^{2} and 4x^{2} to get 10x^{2}.

$$2x^{3}+10x^{2}+6x+2+8x+4-3\left(x+1\right)+6$$

Combine 6x and 8x to get 14x.

$$2x^{3}+10x^{2}+14x+2+4-3\left(x+1\right)+6$$

Add 2 and 4 to get 6.

$$2x^{3}+10x^{2}+14x+6-3\left(x+1\right)+6$$

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

$$2x^{3}+10x^{2}+14x+6-3x-3+6$$

Combine 14x and -3x to get 11x.

$$2x^{3}+10x^{2}+11x+6-3+6$$

Subtract 3 from 6 to get 3.

$$2x^{3}+10x^{2}+11x+3+6$$

Add 3 and 6 to get 9.

$$2x^{3}+10x^{2}+11x+9$$