Solve 4x^3+25x^2+25x

$$4 x ^ { 3 } + 25 x ^ { 2 } + 25 x$$

Factor

$$x\left(x+5\right)\left(4x+5\right)$$

Solution Steps

Factor out x.

$$x\left(4x^{2}+25x+25\right)$$

Consider 4x^{2}+25x+25. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+25. To find a and b, set up a system to be solved.

$$a+b=25 ab=4\times 25=100$$

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 100.

$$1,100 2,50 4,25 5,20 10,10$$

Calculate the sum for each pair.

$$1+100=101 2+50=52 4+25=29 5+20=25 10+10=20$$

The solution is the pair that gives sum 25.

$$a=5 b=20$$

Rewrite 4x^{2}+25x+25 as \left(4x^{2}+5x\right)+\left(20x+25\right).

$$\left(4x^{2}+5x\right)+\left(20x+25\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(4x+5\right)+5\left(4x+5\right)$$

Factor out common term 4x+5 by using distributive property.

$$\left(4x+5\right)\left(x+5\right)$$

Rewrite the complete factored expression.

$$x\left(4x+5\right)\left(x+5\right)$$

Evaluate

$$x\left(x+5\right)\left(4x+5\right)$$