Solve f(x)=(x+4)^2(x-3)^2(x-1)

$$f ( x ) = ( x + 4 ) ^ { 2 } ( x - 3 ) ^ { 2 } ( x - 1 )$$

Evaluate

$$\left(x-1\right)\left(\left(x-3\right)\left(x+4\right)\right)^{2}$$

Solution Steps

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

$$\left(x^{2}+8x+16\right)\left(x-3\right)^{2}\left(x-1\right)$$

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

$$\left(x^{2}+8x+16\right)\left(x^{2}-6x+9\right)\left(x-1\right)$$

Use the distributive property to multiply x^{2}+8x+16 by x^{2}-6x+9 and combine like terms.

$$\left(x^{4}+2x^{3}-23x^{2}-24x+144\right)\left(x-1\right)$$

Use the distributive property to multiply x^{4}+2x^{3}-23x^{2}-24x+144 by x-1 and combine like terms.

$$x^{5}+x^{4}-25x^{3}-x^{2}+168x-144$$

Expand

$$x^{5}+x^{4}-25x^{3}-x^{2}+168x-144$$

Solution Steps

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

$$\left(x^{2}+8x+16\right)\left(x-3\right)^{2}\left(x-1\right)$$

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

$$\left(x^{2}+8x+16\right)\left(x^{2}-6x+9\right)\left(x-1\right)$$

Use the distributive property to multiply x^{2}+8x+16 by x^{2}-6x+9 and combine like terms.

$$\left(x^{4}+2x^{3}-23x^{2}-24x+144\right)\left(x-1\right)$$

Use the distributive property to multiply x^{4}+2x^{3}-23x^{2}-24x+144 by x-1 and combine like terms.

$$x^{5}+x^{4}-25x^{3}-x^{2}+168x-144$$