Solve 4x^3+15x^2+14x

$$4 x ^ { 3 } + 15 x ^ { 2 } + 14 x$$

Factor

$$x\left(x+2\right)\left(4x+7\right)$$

Solution Steps

Factor out x.

$$x\left(4x^{2}+15x+14\right)$$

Consider 4x^{2}+15x+14. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+14. To find a and b, set up a system to be solved.

$$a+b=15 ab=4\times 14=56$$

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 56.

$$1,56 2,28 4,14 7,8$$

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 15.

$$a=7 b=8$$

Rewrite 4x^{2}+15x+14 as \left(4x^{2}+7x\right)+\left(8x+14\right).

$$\left(4x^{2}+7x\right)+\left(8x+14\right)$$

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

$$x\left(4x+7\right)+2\left(4x+7\right)$$

Factor out common term 4x+7 by using distributive property.

$$\left(4x+7\right)\left(x+2\right)$$

Rewrite the complete factored expression.

$$x\left(4x+7\right)\left(x+2\right)$$

Evaluate

$$x\left(x+2\right)\left(4x+7\right)$$