Factorize:

We have:

$6 x^{2}-5 x-21$

We have to split $(-5)$ into two numbers such that their sum is $(-5)$ and their product is $(-126)$, i.e., $6 \times(-21)$.

Clearly, $9+(-14)=-5$ and $9 \times(-14)=-126$.

$\therefore 6 x^{2}-5 x-21=6 x^{2}+9 x-14 x-21$

$=3 x(2 x+3)-7(2 x+3)$

$=(2 x+3)(3 x-7)$