Solve x^2(2-x)^2+(x-6)(x+1)-x^3-27-3(x-1)(x+1)

$$x ^ { 2 } ( 2 - x ) ^ { 2 } + ( x - 6 ) ( x + 1 ) - x ^ { 3 } - 27 - 3 ( x - 1 ) ( x + 1 )$$

Evaluate

$$x^{4}-5x^{3}+2x^{2}-5x-30$$

Solution Steps

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

$$x^{2}\left(4-4x+x^{2}\right)+\left(x-6\right)\left(x+1\right)-x^{3}-27-3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x^{2} by 4-4x+x^{2}.

$$4x^{2}-4x^{3}+x^{4}+\left(x-6\right)\left(x+1\right)-x^{3}-27-3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-6 by x+1 and combine like terms.

$$4x^{2}-4x^{3}+x^{4}+x^{2}-5x-6-x^{3}-27-3\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-4x^{3}+x^{4}-5x-6-x^{3}-27-3\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-5x^{3}+x^{4}-5x-6-27-3\left(x-1\right)\left(x+1\right)$$

Subtract 27 from -6 to get -33.

$$5x^{2}-5x^{3}+x^{4}-5x-33-3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply -3 by x-1.

$$5x^{2}-5x^{3}+x^{4}-5x-33+\left(-3x+3\right)\left(x+1\right)$$

Use the distributive property to multiply -3x+3 by x+1 and combine like terms.

$$5x^{2}-5x^{3}+x^{4}-5x-33-3x^{2}+3$$

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

$$2x^{2}-5x^{3}+x^{4}-5x-33+3$$

Add -33 and 3 to get -30.

$$2x^{2}-5x^{3}+x^{4}-5x-30$$

Expand

$$x^{4}-5x^{3}+2x^{2}-5x-30$$

Solution Steps

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

$$x^{2}\left(4-4x+x^{2}\right)+\left(x-6\right)\left(x+1\right)-x^{3}-27-3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x^{2} by 4-4x+x^{2}.

$$4x^{2}-4x^{3}+x^{4}+\left(x-6\right)\left(x+1\right)-x^{3}-27-3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply x-6 by x+1 and combine like terms.

$$4x^{2}-4x^{3}+x^{4}+x^{2}-5x-6-x^{3}-27-3\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-4x^{3}+x^{4}-5x-6-x^{3}-27-3\left(x-1\right)\left(x+1\right)$$

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

$$5x^{2}-5x^{3}+x^{4}-5x-6-27-3\left(x-1\right)\left(x+1\right)$$

Subtract 27 from -6 to get -33.

$$5x^{2}-5x^{3}+x^{4}-5x-33-3\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply -3 by x-1.

$$5x^{2}-5x^{3}+x^{4}-5x-33+\left(-3x+3\right)\left(x+1\right)$$

Use the distributive property to multiply -3x+3 by x+1 and combine like terms.

$$5x^{2}-5x^{3}+x^{4}-5x-33-3x^{2}+3$$

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

$$2x^{2}-5x^{3}+x^{4}-5x-33+3$$

Add -33 and 3 to get -30.

$$2x^{2}-5x^{3}+x^{4}-5x-30$$