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

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

Evaluate

$$x^{2}+6x+2$$

Solution Steps

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

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

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

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

Use the distributive property to multiply -2 by x^{2}-2x+1.

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

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

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

Combine 4x and 4x to get 8x.

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

Subtract 2 from 1 to get -1.

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

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

$$2x^{2}+8x-1-2x+3-x^{2}$$

Combine 8x and -2x to get 6x.

$$2x^{2}+6x-1+3-x^{2}$$

Add -1 and 3 to get 2.

$$2x^{2}+6x+2-x^{2}$$

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

$$x^{2}+6x+2$$

Expand

$$x^{2}+6x+2$$

Solution Steps

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

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

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

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

Use the distributive property to multiply -2 by x^{2}-2x+1.

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

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

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

Combine 4x and 4x to get 8x.

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

Subtract 2 from 1 to get -1.

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

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

$$2x^{2}+8x-1-2x+3-x^{2}$$

Combine 8x and -2x to get 6x.

$$2x^{2}+6x-1+3-x^{2}$$

Add -1 and 3 to get 2.

$$2x^{2}+6x+2-x^{2}$$

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

$$x^{2}+6x+2$$