Solve 30w^3+76w^2+14w

$$30 w ^ { 3 } + 76 w ^ { 2 } + 14 w$$

Factor

$$2w\left(3w+7\right)\left(5w+1\right)$$

Solution Steps

Factor out 2.

$$2\left(15w^{3}+38w^{2}+7w\right)$$

Consider 15w^{3}+38w^{2}+7w. Factor out w.

$$w\left(15w^{2}+38w+7\right)$$

Consider 15w^{2}+38w+7. Factor the expression by grouping. First, the expression needs to be rewritten as 15w^{2}+aw+bw+7. To find a and b, set up a system to be solved.

$$a+b=38 ab=15\times 7=105$$

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 105.

$$1,105 3,35 5,21 7,15$$

Calculate the sum for each pair.

$$1+105=106 3+35=38 5+21=26 7+15=22$$

The solution is the pair that gives sum 38.

$$a=3 b=35$$

Rewrite 15w^{2}+38w+7 as \left(15w^{2}+3w\right)+\left(35w+7\right).

$$\left(15w^{2}+3w\right)+\left(35w+7\right)$$

Factor out 3w in the first and 7 in the second group.

$$3w\left(5w+1\right)+7\left(5w+1\right)$$

Factor out common term 5w+1 by using distributive property.

$$\left(5w+1\right)\left(3w+7\right)$$

Rewrite the complete factored expression.

$$2w\left(5w+1\right)\left(3w+7\right)$$

Evaluate

$$2w\left(3w+7\right)\left(5w+1\right)$$