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

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

Evaluate

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

Solution Steps

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

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

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

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

Multiply -1 and -1 to get 1.

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Combine -x and -x to get 2\left(-x\right).

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Multiply 2 and -1 to get -2.

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

Combine x^{3}\left(-1\right) and x^{3} to get 0.

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

Expand

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

Solution Steps

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

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

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

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

Multiply -1 and -1 to get 1.

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Combine -x and -x to get 2\left(-x\right).

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Multiply 2 and -1 to get -2.

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

Combine x^{3}\left(-1\right) and x^{3} to get 0.

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