Solve 10x^4+15x^3+5x^2

$$10 x ^ { 4 } + 15 x ^ { 3 } + 5 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out 5.

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

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

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

Consider 2x^{2}+3x+1. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=3 ab=2\times 1=2$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=2$$

Rewrite 2x^{2}+3x+1 as \left(2x^{2}+x\right)+\left(2x+1\right).

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

Factor out x in 2x^{2}+x.

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

Factor out common term 2x+1 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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