Solve x^4+2x^3+x^2+2x

$$x ^ { 4 } + 2 x ^ { 3 } + x ^ { 2 } + 2 x$$

Factor

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

Solution Steps

Factor out x.

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

Consider x^{3}+2x^{2}+x+2. Do the grouping x^{3}+2x^{2}+x+2=\left(x^{3}+2x^{2}\right)+\left(x+2\right), and factor out x^{2} in x^{3}+2x^{2}.

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

Factor out common term x+2 by using distributive property.

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

Rewrite the complete factored expression. Polynomial x^{2}+1 is not factored since it does not have any rational roots.

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

Evaluate

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