Factorize the polynomial expression $36a^2+36a+9$, and choose the option which represents the factored form correctly.

The given expression is as follows, 
$36a^2+36a+9$.
It can be observed that 9 can be factored out from each of the three terms. So, taking 3 as common, we can rewrite the expression as follows,
$9\left(4a^2+4a+1\right)$
Now, to factorize the above expression further, we need to take the product of the first and last terms of the above expression, i.e., 
$4a^2\times 1=4a^2$
Next, we need to split middle term of the polynomial expression, i.e., $4a$as a sum or difference of two terms, such that their product is equal to the product of first and last terms, i.e., $4a^2$. 
$9\left(4a^2+4a+1\right)\iff 9\left(4a^2+2a+2a+1\right)$
Now, we need to group the terms and factor out the common factors, as follows,
$\begin{array}{c}9\left(4a^2+2a\right)+\left(2a+1\right)=9\left\{2a\left(2a+1\right)+1\left(2a+1\right)\right\}\\ =9\left(2a+1\right)\left(2a+1\right)\\ =9{\left(2a+1\right)}^2\end{array}$
So, the factored form of the given polynomial expression is $9{\left(2a+1\right)}^2$.