Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
$$-6a^{2}+5a+6$$
Factor the expression by grouping. First, the expression needs to be rewritten as $-6a^{2}+pa+qa+6$. To find $p$ and $q$, set up a system to be solved.
$$p+q=5$$ $$pq=-6\times 6=-36$$
Since $pq$ is negative, $p$ and $q$ have the opposite signs. Since $p+q$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-36$.
Rewrite $-6a^{2}+5a+6$ as $\left(-6a^{2}+9a\right)+\left(-4a+6\right)$.
$$\left(-6a^{2}+9a\right)+\left(-4a+6\right)$$
Factor out $-3a$ in the first and $-2$ in the second group.
$$-3a\left(2a-3\right)-2\left(2a-3\right)$$
Factor out common term $2a-3$ by using distributive property.
$$\left(2a-3\right)\left(-3a-2\right)$$
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 $ax^{2}+bx+c=0$.
$$-6a^{2}+5a+6=0$$
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 $a=\frac{-5±13}{-12}$ when $±$ is plus. Add $-5$ to $13$.
$$a=\frac{8}{-12}$$
Reduce the fraction $\frac{8}{-12}$ to lowest terms by extracting and canceling out $4$.
$$a=-\frac{2}{3}$$
Now solve the equation $a=\frac{-5±13}{-12}$ when $±$ is minus. Subtract $13$ from $-5$.
$$a=-\frac{18}{-12}$$
Reduce the fraction $\frac{-18}{-12}$ to lowest terms by extracting and canceling out $6$.
$$a=\frac{3}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-\frac{2}{3}$ for $x_{1}$ and $\frac{3}{2}$ for $x_{2}$.
Multiply $\frac{-3a-2}{-3}$ times $\frac{-2a+3}{-2}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
