Solve (12x^2-6x-5)+(-4x^2+7x-2)

$$( 12 x ^ { 2 } - 6 x - 5 ) + ( - 4 x ^ { 2 } + 7 x - 2 )$$

Evaluate

$$\left(8x-7\right)\left(x+1\right)$$

Solution Steps

Combine 12x^{2} and -4x^{2} to get 8x^{2}.

$$8x^{2}-6x-5+7x-2$$

Combine -6x and 7x to get x.

$$8x^{2}+x-5-2$$

Subtract 2 from -5 to get -7.

$$8x^{2}+x-7$$

Factor

$$\left(8x-7\right)\left(x+1\right)$$

Solution Steps

Multiply and combine like terms.

$$8x^{2}+x-7$$

Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=1 ab=8\left(-7\right)=-56$$

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. List all such integer pairs that give product -56.

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

Calculate the sum for each pair.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

The solution is the pair that gives sum 1.

$$a=-7 b=8$$

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

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

Factor out x in 8x^{2}-7x.

$$x\left(8x-7\right)+8x-7$$

Factor out common term 8x-7 by using distributive property.

$$\left(8x-7\right)\left(x+1\right)$$