Factor the expression by grouping. First, the expression needs to be rewritten as $8x^{2}+ax+bx-5$. To find $a$ and $b$, set up a system to be solved.
$$a+b=18$$ $$ab=8\left(-5\right)=-40$$
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 $-40$.
$$-1,40$$ $$-2,20$$ $$-4,10$$ $$-5,8$$
Calculate the sum for each pair.
$$-1+40=39$$ $$-2+20=18$$ $$-4+10=6$$ $$-5+8=3$$
The solution is the pair that gives sum $18$.
$$a=-2$$ $$b=20$$
Rewrite $8x^{2}+18x-5$ as $\left(8x^{2}-2x\right)+\left(20x-5\right)$.
$$\left(8x^{2}-2x\right)+\left(20x-5\right)$$
Factor out $2x$ in the first and $5$ in the second group.
$$2x\left(4x-1\right)+5\left(4x-1\right)$$
Factor out common term $4x-1$ by using distributive property.
$$\left(4x-1\right)\left(2x+5\right)$$
Steps Using the Quadratic Formula
Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$8x^{2}+18x-5=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Now solve the equation $x=\frac{-18±22}{16}$ when $±$ is plus. Add $-18$ to $22$.
$$x=\frac{4}{16}$$
Reduce the fraction $\frac{4}{16}$ to lowest terms by extracting and canceling out $4$.
$$x=\frac{1}{4}$$
Now solve the equation $x=\frac{-18±22}{16}$ when $±$ is minus. Subtract $22$ from $-18$.
$$x=-\frac{40}{16}$$
Reduce the fraction $\frac{-40}{16}$ to lowest terms by extracting and canceling out $8$.
$$x=-\frac{5}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{1}{4}$ for $x_{1}$ and $-\frac{5}{2}$ for $x_{2}$.
Multiply $\frac{4x-1}{4}$ times $\frac{2x+5}{2}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
