Solve (x^2+x+1)(-x^4-x^3+2x)

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

Evaluate

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

Solution Steps

Use the distributive property to multiply x^{2}+x+1 by -x^{4}-x^{3}+2x and combine like terms.

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

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

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

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

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

Combine -x^{5} and x^{5}\left(-1\right) to get -2x^{5}.

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

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

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

Expand

$$2x+2x^{2}+x^{3}-2x^{4}-2x^{5}-x^{6}$$

Solution Steps

Use the distributive property to multiply x^{2}+x+1 by -x^{4}-x^{3}+2x and combine like terms.

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

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

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

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

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

Combine -x^{5} and x^{5}\left(-1\right) to get -2x^{5}.

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

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

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