Evaluate the expression:
$2a + 6b - 3{(a + 3b)^2}$

Hint: We will solve the arithmetic equations by first solving the brackets and then by applying the BODMAS rule, will solve it further. BODMAS rule is the order in which we solve arithmetic equations.

Complete step-by-step solution:
The equation is $2a + 6b - 3{(a + 3b)^2}$
Taking $2$ common from the first two terms i.e., $2a + 6b$ ,
We get the equation as, $2(a + 3b) - 3{(a + 3b)^2}$
We see that $(a + 3b)$ is common in both the terms of the equation.
So, taking $(a + 3b)$ common from the equation, we get the equation as,
$(a + 3b)(2 - 3(a + 3b))$
Multiplying $3$ with $(a + 3b)$, we get the equation as,
$(a + 3b)(2 - 3a - 9b)$
Simplifying further and multiplying the brackets, we get the equation as,
$a(2 - 3a - 9b) + 3b(2 - 3a - 9b)$
Multiplying the brackets, we get the equation as.
$2a - 3{a^2} - 9ab + 6b - 9ab - 27{b^2}$
Taking similar terms together, we get the equation as,
$ - 3{a^2} - 27{b^2} - 9ab - 9ab + 2a + 6b$
Solving the similar terms, we get the equation as,
$ - 3{a^2} - 27{b^2} - 18ab + 2a + 6b$
Therefore, we get the solution as $2a + 6b - 3{(a + 3b)^2} = - 3{a^2} - 27{b^2} - 18ab + 2a + 6b$.

Note: BODMAS full form is Brackets, of, Division, Multiplication, Addition, and Subtraction. BODMAS is sometimes referred to as PEDMAS - Parentheses, Exponents, Division, Multiplication, Addition, and Subtraction. If there are no brackets or parentheses, we start to solve the equation by solving the division part and the Multiplication, Addition, and Subtraction.