Solve f(x)=(x+5)^2(x-2)(x-6)^2

$$f ( x ) = ( x + 5 ) ^ { 2 } ( x - 2 ) ( x - 6 ) ^ { 2 }$$

Evaluate

$$\left(x-2\right)\left(\left(x-6\right)\left(x+5\right)\right)^{2}$$

Solution Steps

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

$$\left(x^{2}+10x+25\right)\left(x-2\right)\left(x-6\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$\left(x^{2}+10x+25\right)\left(x-2\right)\left(x^{2}-12x+36\right)$$

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

$$\left(x^{3}+8x^{2}+5x-50\right)\left(x^{2}-12x+36\right)$$

Use the distributive property to multiply x^{3}+8x^{2}+5x-50 by x^{2}-12x+36 and combine like terms.

$$x^{5}-4x^{4}-55x^{3}+178x^{2}+780x-1800$$

Expand

$$x^{5}-4x^{4}-55x^{3}+178x^{2}+780x-1800$$

Solution Steps

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

$$\left(x^{2}+10x+25\right)\left(x-2\right)\left(x-6\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$\left(x^{2}+10x+25\right)\left(x-2\right)\left(x^{2}-12x+36\right)$$

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

$$\left(x^{3}+8x^{2}+5x-50\right)\left(x^{2}-12x+36\right)$$

Use the distributive property to multiply x^{3}+8x^{2}+5x-50 by x^{2}-12x+36 and combine like terms.

$$x^{5}-4x^{4}-55x^{3}+178x^{2}+780x-1800$$