Factor the expression by grouping. First, the expression needs to be rewritten as $8x^{2}+ax+bx-15$. To find $a$ and $b$, set up a system to be solved.
$$a+b=2$$ $$ab=8\left(-15\right)=-120$$
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 $-120$.
Rewrite $8x^{2}+2x-15$ as $\left(8x^{2}-10x\right)+\left(12x-15\right)$.
$$\left(8x^{2}-10x\right)+\left(12x-15\right)$$
Factor out $2x$ in the first and $3$ in the second group.
$$2x\left(4x-5\right)+3\left(4x-5\right)$$
Factor out common term $4x-5$ by using distributive property.
$$\left(4x-5\right)\left(2x+3\right)$$
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 $ax^{2}+bx+c=0$.
$$8x^{2}+2x-15=0$$
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 $x=\frac{-2±22}{16}$ when $±$ is plus. Add $-2$ to $22$.
$$x=\frac{20}{16}$$
Reduce the fraction $\frac{20}{16}$ to lowest terms by extracting and canceling out $4$.
$$x=\frac{5}{4}$$
Now solve the equation $x=\frac{-2±22}{16}$ when $±$ is minus. Subtract $22$ from $-2$.
$$x=-\frac{24}{16}$$
Reduce the fraction $\frac{-24}{16}$ to lowest terms by extracting and canceling out $8$.
$$x=-\frac{3}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{5}{4}$ for $x_{1}$ and $-\frac{3}{2}$ for $x_{2}$.
Multiply $\frac{4x-5}{4}$ times $\frac{2x+3}{2}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form $x^2+Bx+C=0$.This is achieved by dividing both sides of the equation by $8$
$$x ^ 2 +\frac{1}{4}x -\frac{15}{8} = 0$$
Let $r$ and $s$ be the factors for the quadratic equation such that $x^2+Bx+C=(x−r)(x−s)$ where sum of factors $(r+s)=−B$ and the product of factors $rs = C$
$$r + s = -\frac{1}{4} $$ $$ rs = -\frac{15}{8}$$
Two numbers $r$ and $s$ sum up to $-\frac{1}{4}$ exactly when the average of the two numbers is $\frac{1}{2}*-\frac{1}{4} = -\frac{1}{8}$. You can also see that the midpoint of $r$ and $s$ corresponds to the axis of symmetry of the parabola represented by the quadratic equation $y=x^2+Bx+C$. The values of $r$ and $s$ are equidistant from the center by an unknown quantity $u$. Express $r$ and $s$ with respect to variable $u$.
$$r = -\frac{1}{8} - u$$ $$s = -\frac{1}{8} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -\frac{15}{8}$
