# How do you factor 6x^2+x-15?

May 1, 2015

$6 {x}^{2} + x - 15$

Base factors (without regard to signs) of
$6 = {S}_{6} = \left\{\begin{matrix}1 \times 6 \\ 2 \times 3\end{matrix}\right\}$

Base factors (without regard to signs) of
$15 = {S}_{15} = \left\{\begin{matrix}1 \times 15 \\ 3 \times 5\end{matrix}\right\}$

Since in the given expression the term $15$ is negative
we are looking for a pair from ${S}_{6}$ and another pair from ${S}_{15}$
that can be multiplied as
one term from ${S}_{6}$ times one term from ${S}_{15}$
subtracted from
the other term from ${S}_{6}$ times the other term from ${S}_{15}$
with a result of $1$ (the coefficient of the $x$ term).

Only a little thinking produces the pairs:
$\textcolor{red}{\text{("3,2")") " and " color(blue)("("3,5")}}$

and the solution quickly follows
$6 {x}^{2} + x - 15$

$= \left(\textcolor{red}{3} x + \textcolor{b l u e}{5}\right) \left(\textcolor{red}{2} x - \textcolor{b l u e}{3}\right)$