Factor the expression by grouping. First, the expression needs to be rewritten as $8n^{2}+an+bn-3$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-10$$ $$ab=8\left(-3\right)=-24$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product $-24$.
$$1,-24$$ $$2,-12$$ $$3,-8$$ $$4,-6$$
Calculate the sum for each pair.
$$1-24=-23$$ $$2-12=-10$$ $$3-8=-5$$ $$4-6=-2$$
The solution is the pair that gives sum $-10$.
$$a=-12$$ $$b=2$$
Rewrite $8n^{2}-10n-3$ as $\left(8n^{2}-12n\right)+\left(2n-3\right)$.
$$\left(8n^{2}-12n\right)+\left(2n-3\right)$$
Factor out $4n$ in $8n^{2}-12n$.
$$4n\left(2n-3\right)+2n-3$$
Factor out common term $2n-3$ by using distributive property.
$$\left(2n-3\right)\left(4n+1\right)$$
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 $ax^{2}+bx+c=0$.
$$8n^{2}-10n-3=0$$
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 $n=\frac{10±14}{16}$ when $±$ is plus. Add $10$ to $14$.
$$n=\frac{24}{16}$$
Reduce the fraction $\frac{24}{16}$ to lowest terms by extracting and canceling out $8$.
$$n=\frac{3}{2}$$
Now solve the equation $n=\frac{10±14}{16}$ when $±$ is minus. Subtract $14$ from $10$.
$$n=-\frac{4}{16}$$
Reduce the fraction $\frac{-4}{16}$ to lowest terms by extracting and canceling out $4$.
$$n=-\frac{1}{4}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{3}{2}$ for $x_{1}$ and $-\frac{1}{4}$ for $x_{2}$.
Multiply $\frac{2n-3}{2}$ times $\frac{4n+1}{4}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
