Solve x^2(2-2x)(1-x)+2x^2(2x-x^2)+x(1-x)+(1-x)^2

$$x ^ { 2 } ( 2 - 2 x ) ( 1 - x ) + 2 x ^ { 2 } ( 2 x - x ^ { 2 } ) + x ( 1 - x ) + ( 1 - x ) ^ { 2 }$$

Evaluate

$$2x^{2}-x+1$$

Solution Steps

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

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

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

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

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

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

Combine -4x^{3} and 4x^{3} to get 0.

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

Combine 2x^{4} and -2x^{4} to get 0.

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

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

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

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

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

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

$$x^{2}+x+1-2x+x^{2}$$

Combine x and -2x to get -x.

$$x^{2}-x+1+x^{2}$$

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

$$2x^{2}-x+1$$

Expand

$$2x^{2}-x+1$$

Solution Steps

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

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

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

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

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

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

Combine -4x^{3} and 4x^{3} to get 0.

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

Combine 2x^{4} and -2x^{4} to get 0.

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

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

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

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

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

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

$$x^{2}+x+1-2x+x^{2}$$

Combine x and -2x to get -x.

$$x^{2}-x+1+x^{2}$$

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

$$2x^{2}-x+1$$