Solve (-4x^2+14x-11)+(12x^2-6x+5)

$$( - 4 x ^ { 2 } + 14 x - 11 ) + ( 12 x ^ { 2 } - 6 x + 5 )$$

Evaluate

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

Solution Steps

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

$$8x^{2}+14x-11-6x+5$$

Combine 14x and -6x to get 8x.

$$8x^{2}+8x-11+5$$

Add -11 and 5 to get -6.

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

Factor

$$2\left(2x-1\right)\left(2x+3\right)$$

Solution Steps

Multiply and combine like terms.

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

Factor out 2.

$$2\left(4x^{2}+4x-3\right)$$

Consider 4x^{2}+4x-3. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=4 ab=4\left(-3\right)=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 4.

$$a=-2 b=6$$

Rewrite 4x^{2}+4x-3 as \left(4x^{2}-2x\right)+\left(6x-3\right).

$$\left(4x^{2}-2x\right)+\left(6x-3\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(2x-1\right)+3\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(2x+3\right)$$

Rewrite the complete factored expression.

$$2\left(2x-1\right)\left(2x+3\right)$$