Solve 2x+3x^2-6+x-5x^2+5

$$2 x + 3 x ^ { 2 } - 6 + x - 5 x ^ { 2 } + 5$$

Evaluate

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

Solution Steps

Combine 2x and x to get 3x.

$$3x+3x^{2}-6-5x^{2}+5$$

Combine 3x^{2} and -5x^{2} to get -2x^{2}.

$$3x-2x^{2}-6+5$$

Add -6 and 5 to get -1.

$$3x-2x^{2}-1$$

Factor

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

Solution Steps

Multiply and combine like terms.

$$-2x^{2}+3x-1$$

Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=3 ab=-2\left(-1\right)=2$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=2 b=1$$

Rewrite -2x^{2}+3x-1 as \left(-2x^{2}+2x\right)+\left(x-1\right).

$$\left(-2x^{2}+2x\right)+\left(x-1\right)$$

Factor out 2x in the first and -1 in the second group.

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

Factor out common term -x+1 by using distributive property.

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