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

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

Evaluate

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

Solution Steps

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

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

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

$$9+6x+x^{2}-\left(27+27x+9x^{2}+x^{3}\right)$$

To find the opposite of 27+27x+9x^{2}+x^{3}, find the opposite of each term.

$$9+6x+x^{2}-27-27x-9x^{2}-x^{3}$$

Subtract 27 from 9 to get -18.

$$-18+6x+x^{2}-27x-9x^{2}-x^{3}$$

Combine 6x and -27x to get -21x.

$$-18-21x+x^{2}-9x^{2}-x^{3}$$

Combine x^{2} and -9x^{2} to get -8x^{2}.

$$-18-21x-8x^{2}-x^{3}$$

Expand

$$-x^{3}-8x^{2}-21x-18$$

Solution Steps

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

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

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

$$9+6x+x^{2}-\left(27+27x+9x^{2}+x^{3}\right)$$

To find the opposite of 27+27x+9x^{2}+x^{3}, find the opposite of each term.

$$9+6x+x^{2}-27-27x-9x^{2}-x^{3}$$

Subtract 27 from 9 to get -18.

$$-18+6x+x^{2}-27x-9x^{2}-x^{3}$$

Combine 6x and -27x to get -21x.

$$-18-21x+x^{2}-9x^{2}-x^{3}$$

Combine x^{2} and -9x^{2} to get -8x^{2}.

$$-18-21x-8x^{2}-x^{3}$$