Factorize each of the following algebraic expression:

To factorise $\mathrm{x}^{2}+12 \mathrm{x}-45$, we will find two numbers $\mathrm{p}$ and $\mathrm{q}$ such that $\mathrm{p}+\mathrm{q}=12$ and $\mathrm{pq}=-45$.

Now

$15+(-3)=12$

and

$15 \times(-3)=-45$

Splitting the middle term $12 \mathrm{x}$ in the given quadratic as $-3 \mathrm{x}+15 \mathrm{x}$, we get:

$x^{2}+12 x-45=x^{2}-3 x+15 x-45$

$=\left(\mathrm{x}^{2}-3 \mathrm{x}\right)+(15 \mathrm{x}-45)$

$=\mathrm{x}(\mathrm{x}-3)+15(\mathrm{x}-3)$

$=(\mathrm{x}+15)(\mathrm{x}-3)$