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

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

Evaluate

$$2\left(x+1\right)$$

Solution Steps

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

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

Calculate -x to the power of 2 and get x^{2}.

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

Multiply -2 and -1 to get 2.

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

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

To find the opposite of x^{2}-1, find the opposite of each term.

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

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

$$2x+1+1$$

Add 1 and 1 to get 2.

$$2x+2$$

Expand

$$2x+2$$

Solution Steps

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

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

Calculate -x to the power of 2 and get x^{2}.

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

Multiply -2 and -1 to get 2.

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

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

To find the opposite of x^{2}-1, find the opposite of each term.

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

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

$$2x+1+1$$

Add 1 and 1 to get 2.

$$2x+2$$