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

$$f ( x ) = - ( x + 2 ) ( x + 2 ) ( x - 1 ) ( x - 1 ) ( x + 3 )$$

Evaluate

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

Solution Steps

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

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

To find the opposite of x+2, find the opposite of each term.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

Apply the distributive property by multiplying each term of -x-2 by each term of x+2.

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

Combine -2x and -2x to get -4x.

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

Apply the distributive property by multiplying each term of -x^{2}-4x-4 by each term of x^{2}-2x+1.

$$\left(-x^{4}+2x^{3}-x^{2}-4x^{3}+8x^{2}-4x-4x^{2}+8x-4\right)\left(x+3\right)$$

Combine 2x^{3} and -4x^{3} to get -2x^{3}.

$$\left(-x^{4}-2x^{3}-x^{2}+8x^{2}-4x-4x^{2}+8x-4\right)\left(x+3\right)$$

Combine -x^{2} and 8x^{2} to get 7x^{2}.

$$\left(-x^{4}-2x^{3}+7x^{2}-4x-4x^{2}+8x-4\right)\left(x+3\right)$$

Combine 7x^{2} and -4x^{2} to get 3x^{2}.

$$\left(-x^{4}-2x^{3}+3x^{2}-4x+8x-4\right)\left(x+3\right)$$

Combine -4x and 8x to get 4x.

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

Apply the distributive property by multiplying each term of -x^{4}-2x^{3}+3x^{2}+4x-4 by each term of x+3.

$$-x^{5}-3x^{4}-2x^{4}-6x^{3}+3x^{3}+9x^{2}+4x^{2}+12x-4x-12$$

Combine -3x^{4} and -2x^{4} to get -5x^{4}.

$$-x^{5}-5x^{4}-6x^{3}+3x^{3}+9x^{2}+4x^{2}+12x-4x-12$$

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

$$-x^{5}-5x^{4}-3x^{3}+9x^{2}+4x^{2}+12x-4x-12$$

Combine 9x^{2} and 4x^{2} to get 13x^{2}.

$$-x^{5}-5x^{4}-3x^{3}+13x^{2}+12x-4x-12$$

Combine 12x and -4x to get 8x.

$$-x^{5}-5x^{4}-3x^{3}+13x^{2}+8x-12$$

Expand

$$-\left(x^{5}+5x^{4}+3x^{3}-13x^{2}-8x+12\right)$$

Solution Steps

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

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

To find the opposite of x+2, find the opposite of each term.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

Apply the distributive property by multiplying each term of -x-2 by each term of x+2.

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

Combine -2x and -2x to get -4x.

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

Apply the distributive property by multiplying each term of -x^{2}-4x-4 by each term of x^{2}-2x+1.

$$\left(-x^{4}+2x^{3}-x^{2}-4x^{3}+8x^{2}-4x-4x^{2}+8x-4\right)\left(x+3\right)$$

Combine 2x^{3} and -4x^{3} to get -2x^{3}.

$$\left(-x^{4}-2x^{3}-x^{2}+8x^{2}-4x-4x^{2}+8x-4\right)\left(x+3\right)$$

Combine -x^{2} and 8x^{2} to get 7x^{2}.

$$\left(-x^{4}-2x^{3}+7x^{2}-4x-4x^{2}+8x-4\right)\left(x+3\right)$$

Combine 7x^{2} and -4x^{2} to get 3x^{2}.

$$\left(-x^{4}-2x^{3}+3x^{2}-4x+8x-4\right)\left(x+3\right)$$

Combine -4x and 8x to get 4x.

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

Apply the distributive property by multiplying each term of -x^{4}-2x^{3}+3x^{2}+4x-4 by each term of x+3.

$$-x^{5}-3x^{4}-2x^{4}-6x^{3}+3x^{3}+9x^{2}+4x^{2}+12x-4x-12$$

Combine -3x^{4} and -2x^{4} to get -5x^{4}.

$$-x^{5}-5x^{4}-6x^{3}+3x^{3}+9x^{2}+4x^{2}+12x-4x-12$$

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

$$-x^{5}-5x^{4}-3x^{3}+9x^{2}+4x^{2}+12x-4x-12$$

Combine 9x^{2} and 4x^{2} to get 13x^{2}.

$$-x^{5}-5x^{4}-3x^{3}+13x^{2}+12x-4x-12$$

Combine 12x and -4x to get 8x.

$$-x^{5}-5x^{4}-3x^{3}+13x^{2}+8x-12$$