Solve 2x^2+4x-2x^3

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

Factor

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

Solution Steps

Factor out 2.

$$2\left(x^{2}+2x-x^{3}\right)$$

Consider x^{2}+2x-x^{3}. Factor out x.

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

Consider x+2-x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+x+2$$

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

$$a+b=1 ab=-2=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=2 b=-1$$

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

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

Factor out -x in the first and -1 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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