Solve 6x^2+7x-9--5x^2+x-10

$$6 { x }^{ 2 } +7x-9--5 { x }^{ 2 } +x-10$$

Evaluate

$$\left(x-1\right)\left(11x+19\right)$$

Solution Steps

The opposite of -5x^{2} is 5x^{2}.

$$6x^{2}+7x-9+5x^{2}+x-10$$

Combine 6x^{2} and 5x^{2} to get 11x^{2}.

$$11x^{2}+7x-9+x-10$$

Combine 7x and x to get 8x.

$$11x^{2}+8x-9-10$$

Subtract 10 from -9 to get -19.

$$11x^{2}+8x-19$$

Factor

$$\left(x-1\right)\left(11x+19\right)$$

Solution Steps

Multiply and combine like terms.

$$11x^{2}+8x-19$$

Factor the expression by grouping. First, the expression needs to be rewritten as 11x^{2}+ax+bx-19. To find a and b, set up a system to be solved.

$$a+b=8 ab=11\left(-19\right)=-209$$

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 -209.

$$-1,209 -11,19$$

Calculate the sum for each pair.

$$-1+209=208 -11+19=8$$

The solution is the pair that gives sum 8.

$$a=-11 b=19$$

Rewrite 11x^{2}+8x-19 as \left(11x^{2}-11x\right)+\left(19x-19\right).

$$\left(11x^{2}-11x\right)+\left(19x-19\right)$$

Factor out 11x in the first and 19 in the second group.

$$11x\left(x-1\right)+19\left(x-1\right)$$

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

$$\left(x-1\right)\left(11x+19\right)$$