Solve -2x^3-15x^2+38x

$$- 2 x ^ { 3 } - 15 x ^ { 2 } + 38 x$$

Factor

$$x\left(2-x\right)\left(2x+19\right)$$

Solution Steps

Factor out x.

$$x\left(-2x^{2}-15x+38\right)$$

Consider -2x^{2}-15x+38. Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+38. To find a and b, set up a system to be solved.

$$a+b=-15 ab=-2\times 38=-76$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -76.

$$1,-76 2,-38 4,-19$$

Calculate the sum for each pair.

$$1-76=-75 2-38=-36 4-19=-15$$

The solution is the pair that gives sum -15.

$$a=4 b=-19$$

Rewrite -2x^{2}-15x+38 as \left(-2x^{2}+4x\right)+\left(-19x+38\right).

$$\left(-2x^{2}+4x\right)+\left(-19x+38\right)$$

Factor out 2x in the first and 19 in the second group.

$$2x\left(-x+2\right)+19\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(2x+19\right)$$

Rewrite the complete factored expression.

$$x\left(-x+2\right)\left(2x+19\right)$$

Evaluate

$$x\left(2-x\right)\left(2x+19\right)$$