Solve 5left(1+xright)^3+3left(1+xright)^2+2left(1+xright)^2-11=

$$5 { \left(1+x \right) }^{ 3 } +3 { \left(1+x \right) }^{ 2 } +2 { \left(1+x \right) }^{ 2 } -11=$$

Evaluate

$$5\left(x+1\right)^{3}+5\left(x+1\right)^{2}-11$$

Solution Steps

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

$$5\left(1+3x+3x^{2}+x^{3}\right)+3\left(1+x\right)^{2}+2\left(1+x\right)^{2}-11$$

Use the distributive property to multiply 5 by 1+3x+3x^{2}+x^{3}.

$$5+15x+15x^{2}+5x^{3}+3\left(1+x\right)^{2}+2\left(1+x\right)^{2}-11$$

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

$$5+15x+15x^{2}+5x^{3}+3\left(1+2x+x^{2}\right)+2\left(1+x\right)^{2}-11$$

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

$$5+15x+15x^{2}+5x^{3}+3+6x+3x^{2}+2\left(1+x\right)^{2}-11$$

Add 5 and 3 to get 8.

$$8+15x+15x^{2}+5x^{3}+6x+3x^{2}+2\left(1+x\right)^{2}-11$$

Combine 15x and 6x to get 21x.

$$8+21x+15x^{2}+5x^{3}+3x^{2}+2\left(1+x\right)^{2}-11$$

Combine 15x^{2} and 3x^{2} to get 18x^{2}.

$$8+21x+18x^{2}+5x^{3}+2\left(1+x\right)^{2}-11$$

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

$$8+21x+18x^{2}+5x^{3}+2\left(1+2x+x^{2}\right)-11$$

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

$$8+21x+18x^{2}+5x^{3}+2+4x+2x^{2}-11$$

Add 8 and 2 to get 10.

$$10+21x+18x^{2}+5x^{3}+4x+2x^{2}-11$$

Combine 21x and 4x to get 25x.

$$10+25x+18x^{2}+5x^{3}+2x^{2}-11$$

Combine 18x^{2} and 2x^{2} to get 20x^{2}.

$$10+25x+20x^{2}+5x^{3}-11$$

Subtract 11 from 10 to get -1.

$$-1+25x+20x^{2}+5x^{3}$$

Expand

$$5x^{3}+20x^{2}+25x-1$$

Solution Steps

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

$$5\left(1+3x+3x^{2}+x^{3}\right)+3\left(1+x\right)^{2}+2\left(1+x\right)^{2}-11$$

Use the distributive property to multiply 5 by 1+3x+3x^{2}+x^{3}.

$$5+15x+15x^{2}+5x^{3}+3\left(1+x\right)^{2}+2\left(1+x\right)^{2}-11$$

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

$$5+15x+15x^{2}+5x^{3}+3\left(1+2x+x^{2}\right)+2\left(1+x\right)^{2}-11$$

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

$$5+15x+15x^{2}+5x^{3}+3+6x+3x^{2}+2\left(1+x\right)^{2}-11$$

Add 5 and 3 to get 8.

$$8+15x+15x^{2}+5x^{3}+6x+3x^{2}+2\left(1+x\right)^{2}-11$$

Combine 15x and 6x to get 21x.

$$8+21x+15x^{2}+5x^{3}+3x^{2}+2\left(1+x\right)^{2}-11$$

Combine 15x^{2} and 3x^{2} to get 18x^{2}.

$$8+21x+18x^{2}+5x^{3}+2\left(1+x\right)^{2}-11$$

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

$$8+21x+18x^{2}+5x^{3}+2\left(1+2x+x^{2}\right)-11$$

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

$$8+21x+18x^{2}+5x^{3}+2+4x+2x^{2}-11$$

Add 8 and 2 to get 10.

$$10+21x+18x^{2}+5x^{3}+4x+2x^{2}-11$$

Combine 21x and 4x to get 25x.

$$10+25x+18x^{2}+5x^{3}+2x^{2}-11$$

Combine 18x^{2} and 2x^{2} to get 20x^{2}.

$$10+25x+20x^{2}+5x^{3}-11$$

Subtract 11 from 10 to get -1.

$$-1+25x+20x^{2}+5x^{3}$$