Solve 6x^5+15x^4+6x^3+15x^2

$$6 x ^ { 5 } + 15 x ^ { 4 } + 6 x ^ { 3 } + 15 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out 3.

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

Consider 2x^{5}+5x^{4}+2x^{3}+5x^{2}. Factor out x^{2}.

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

Consider 2x^{3}+5x^{2}+2x+5. Do the grouping 2x^{3}+5x^{2}+2x+5=\left(2x^{3}+5x^{2}\right)+\left(2x+5\right), and factor out x^{2} in 2x^{3}+5x^{2}.

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

Factor out common term 2x+5 by using distributive property.

$$\left(2x+5\right)\left(x^{2}+1\right)$$

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

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

Evaluate

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