Solve (x+5)^3-x-5=(x+4)(x+5)(x+6)

$$( x + 5 ) ^ { 3 } - x - 5 = ( x + 4 ) ( x + 5 ) ( x + 6 )$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+5\right)^{3}.

$$x^{3}+15x^{2}+75x+125-x-5=\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Combine 75x and -x to get 74x.

$$x^{3}+15x^{2}+74x+125-5=\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Subtract 5 from 125 to get 120.

$$x^{3}+15x^{2}+74x+120=\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Use the distributive property to multiply x+4 by x+5 and combine like terms.

$$x^{3}+15x^{2}+74x+120=\left(x^{2}+9x+20\right)\left(x+6\right)$$

Use the distributive property to multiply x^{2}+9x+20 by x+6 and combine like terms.

$$x^{3}+15x^{2}+74x+120=x^{3}+15x^{2}+74x+120$$

Subtract x^{3} from both sides.

$$x^{3}+15x^{2}+74x+120-x^{3}=15x^{2}+74x+120$$

Combine x^{3} and -x^{3} to get 0.

$$15x^{2}+74x+120=15x^{2}+74x+120$$

Subtract 15x^{2} from both sides.

$$15x^{2}+74x+120-15x^{2}=74x+120$$

Combine 15x^{2} and -15x^{2} to get 0.

$$74x+120=74x+120$$

Subtract 74x from both sides.

$$74x+120-74x=120$$

Combine 74x and -74x to get 0.

$$120=120$$

Compare 120 and 120.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+5\right)^{3}.

$$x^{3}+15x^{2}+75x+125-x-5=\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Combine 75x and -x to get 74x.

$$x^{3}+15x^{2}+74x+125-5=\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Subtract 5 from 125 to get 120.

$$x^{3}+15x^{2}+74x+120=\left(x+4\right)\left(x+5\right)\left(x+6\right)$$

Use the distributive property to multiply x+4 by x+5 and combine like terms.

$$x^{3}+15x^{2}+74x+120=\left(x^{2}+9x+20\right)\left(x+6\right)$$

Use the distributive property to multiply x^{2}+9x+20 by x+6 and combine like terms.

$$x^{3}+15x^{2}+74x+120=x^{3}+15x^{2}+74x+120$$

Subtract x^{3} from both sides.

$$x^{3}+15x^{2}+74x+120-x^{3}=15x^{2}+74x+120$$

Combine x^{3} and -x^{3} to get 0.

$$15x^{2}+74x+120=15x^{2}+74x+120$$

Subtract 15x^{2} from both sides.

$$15x^{2}+74x+120-15x^{2}=74x+120$$

Combine 15x^{2} and -15x^{2} to get 0.

$$74x+120=74x+120$$

Subtract 74x from both sides.

$$74x+120-74x=120$$

Combine 74x and -74x to get 0.

$$120=120$$

Compare 120 and 120.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$