Solve left(x+4right)^2(x-5)+(x+4)left(x-5right)^2

$${ \left(x+4 \right) }^{ 2 } (x-5)+(x+4) { \left(x-5 \right) }^{ 2 }$$

Evaluate

$$\left(x-5\right)\left(2x-1\right)\left(x+4\right)$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$\left(x^{2}+8x+16\right)\left(x-5\right)+\left(x+4\right)\left(x-5\right)^{2}$$

Use the distributive property to multiply x^{2}+8x+16 by x-5 and combine like terms.

$$x^{3}+3x^{2}-24x-80+\left(x+4\right)\left(x-5\right)^{2}$$

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

$$x^{3}+3x^{2}-24x-80+\left(x+4\right)\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply x+4 by x^{2}-10x+25 and combine like terms.

$$x^{3}+3x^{2}-24x-80+x^{3}-6x^{2}-15x+100$$

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

$$2x^{3}+3x^{2}-24x-80-6x^{2}-15x+100$$

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$2x^{3}-3x^{2}-24x-80-15x+100$$

Combine -24x and -15x to get -39x.

$$2x^{3}-3x^{2}-39x-80+100$$

Add -80 and 100 to get 20.

$$2x^{3}-3x^{2}-39x+20$$

Expand

$$2x^{3}-3x^{2}-39x+20$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$\left(x^{2}+8x+16\right)\left(x-5\right)+\left(x+4\right)\left(x-5\right)^{2}$$

Use the distributive property to multiply x^{2}+8x+16 by x-5 and combine like terms.

$$x^{3}+3x^{2}-24x-80+\left(x+4\right)\left(x-5\right)^{2}$$

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

$$x^{3}+3x^{2}-24x-80+\left(x+4\right)\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply x+4 by x^{2}-10x+25 and combine like terms.

$$x^{3}+3x^{2}-24x-80+x^{3}-6x^{2}-15x+100$$

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

$$2x^{3}+3x^{2}-24x-80-6x^{2}-15x+100$$

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$2x^{3}-3x^{2}-24x-80-15x+100$$

Combine -24x and -15x to get -39x.

$$2x^{3}-3x^{2}-39x-80+100$$

Add -80 and 100 to get 20.

$$2x^{3}-3x^{2}-39x+20$$