Solve (x+4)^3(x-2)+(x+4)^2(x-2)^2

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

Evaluate

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

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+4\right)^{3}.

$$\left(x^{3}+12x^{2}+48x+64\right)\left(x-2\right)+\left(x+4\right)^{2}\left(x-2\right)^{2}$$

Use the distributive property to multiply x^{3}+12x^{2}+48x+64 by x-2 and combine like terms.

$$x^{4}+10x^{3}+24x^{2}-32x-128+\left(x+4\right)^{2}\left(x-2\right)^{2}$$

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

$$x^{4}+10x^{3}+24x^{2}-32x-128+\left(x^{2}+8x+16\right)\left(x-2\right)^{2}$$

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

$$x^{4}+10x^{3}+24x^{2}-32x-128+\left(x^{2}+8x+16\right)\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply x^{2}+8x+16 by x^{2}-4x+4 and combine like terms.

$$x^{4}+10x^{3}+24x^{2}-32x-128+x^{4}+4x^{3}-12x^{2}-32x+64$$

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

$$2x^{4}+10x^{3}+24x^{2}-32x-128+4x^{3}-12x^{2}-32x+64$$

Combine 10x^{3} and 4x^{3} to get 14x^{3}.

$$2x^{4}+14x^{3}+24x^{2}-32x-128-12x^{2}-32x+64$$

Combine 24x^{2} and -12x^{2} to get 12x^{2}.

$$2x^{4}+14x^{3}+12x^{2}-32x-128-32x+64$$

Combine -32x and -32x to get -64x.

$$2x^{4}+14x^{3}+12x^{2}-64x-128+64$$

Add -128 and 64 to get -64.

$$2x^{4}+14x^{3}+12x^{2}-64x-64$$

Expand

$$2x^{4}+14x^{3}+12x^{2}-64x-64$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+4\right)^{3}.

$$\left(x^{3}+12x^{2}+48x+64\right)\left(x-2\right)+\left(x+4\right)^{2}\left(x-2\right)^{2}$$

Use the distributive property to multiply x^{3}+12x^{2}+48x+64 by x-2 and combine like terms.

$$x^{4}+10x^{3}+24x^{2}-32x-128+\left(x+4\right)^{2}\left(x-2\right)^{2}$$

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

$$x^{4}+10x^{3}+24x^{2}-32x-128+\left(x^{2}+8x+16\right)\left(x-2\right)^{2}$$

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

$$x^{4}+10x^{3}+24x^{2}-32x-128+\left(x^{2}+8x+16\right)\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply x^{2}+8x+16 by x^{2}-4x+4 and combine like terms.

$$x^{4}+10x^{3}+24x^{2}-32x-128+x^{4}+4x^{3}-12x^{2}-32x+64$$

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

$$2x^{4}+10x^{3}+24x^{2}-32x-128+4x^{3}-12x^{2}-32x+64$$

Combine 10x^{3} and 4x^{3} to get 14x^{3}.

$$2x^{4}+14x^{3}+24x^{2}-32x-128-12x^{2}-32x+64$$

Combine 24x^{2} and -12x^{2} to get 12x^{2}.

$$2x^{4}+14x^{3}+12x^{2}-32x-128-32x+64$$

Combine -32x and -32x to get -64x.

$$2x^{4}+14x^{3}+12x^{2}-64x-128+64$$

Add -128 and 64 to get -64.

$$2x^{4}+14x^{3}+12x^{2}-64x-64$$