Solve x^2+2x^3+2x^2+2x+2x^2=

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

Evaluate

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

Solution Steps

Combine x^{2} and 2x^{2} to get 3x^{2}.

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

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$5x^{2}+2x^{3}+2x$$

Factor

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

Solution Steps

Factor out x.

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

Consider x+2x^{2}+2x+2+2x. Multiply and combine like terms.

$$2x^{2}+5x+2$$

Consider 2x^{2}+5x+2. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=5 ab=2\times 2=4$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=1 b=4$$

Rewrite 2x^{2}+5x+2 as \left(2x^{2}+x\right)+\left(4x+2\right).

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

Factor out x in the first and 2 in the second group.

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

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

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

Rewrite the complete factored expression.

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