Find the factors of the expression: $2\mathrm{a} + 6\mathrm{b} -3{(\mathrm{a} + 3\mathrm{b})}^2$ 

The given expression:  $2\mathrm{a} + 6\mathrm{b} -3{(\mathrm{a} + 3\mathrm{b})}^2$ .

$\Rightarrow 2\mathrm{a} + 6\mathrm{b} - 3{\left(\mathrm{a} +3\mathrm{b}\right)}^2$

$=2\left(\mathrm{a} + 3\mathrm{b}\right) - 3{\left(\mathrm{a} + 3\mathrm{b}\right)}^2$  

First, we write the prime factors of each term.

Thus, 
$\begin{array}{l}\Rightarrow 2\left(\mathrm{a} + 3\mathrm{b}\right) = 2 \times \left(\mathrm{a} + 3\mathrm{b}\right)\\ \\ \Rightarrow 3{\left(\mathrm{a} + 3\mathrm{b}\right)}^2=3 \times \left(\mathrm{a} + 3\mathrm{b}\right) \times \left(\mathrm{a} + 3\mathrm{b}\right)\end{array}$

 
Now, placing the following factors, we get:  

 $\begin{array}{l}\Rightarrow 2\left(\mathrm{a} + 3\mathrm{b}\right) - 3{\left(\mathrm{a} + 3\mathrm{b}\right)}^2\\ \\ = \left(\mathrm{a} + 3\mathrm{b}\right)\left(2 - 3\left(\mathrm{a} + 3\mathrm{b}\right)\right)\\ \\ = \left(a + 3b\right)\left(2 - 3a - 9b\right)\end{array}$

 
Hence,

The required factors are $\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{3}\mathbf{b}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{-}\mathbf{3}\mathbf{a}\mathbf{-}\mathbf{9}\mathbf{b}\mathbf{)}$ .