Factor the Polynomial Expression: 2a^5 + 8a^6 + 4a^3

We will factorize by extracting the largest common factor. In the given expression, there are the following terms:

$$$2a^5,\hspace{4pt}8a^6,\hspace{4pt}4a^3$

We'll start with the letters, using the law of exponents for multiplication between terms with identical bases, but in reverse order:

$b^{m+n} =b^m\cdot b^n$

to understand that:

$a^6=a^3\cdot a^3\\ a^5=a^3\cdot a^2$

and we'll note that the highest power of $a$that can be extracted as a common factor for all three terms is the power of 3, meaning: $a^3$,

we'll continue and extract the largest common factor for the numbers 2,4,8, which is clearly the number 2 since it is a prime number, therefore we'll conclude and extract the common factor: $2a^3$

We'll apply this to the given expression and extract a common factor:

$2a^5+8a^6+4a^3 =2a^3(a^2+4a^4+2)$

where after extracting the common factor outside the parentheses, we'll look at each term before extracting the common factor separately, asking the question: "By how much did we multiply the common factor to get the current term?" and we'll fill in the missing parts inside the parentheses while making sure that the sign of the term we completed inside the parentheses when multiplied by the sign of the term we extracted outside the parentheses gives us the sign of the original term, it is recommended to verify that the factorization is correct by opening the parentheses, performing the multiplications and confirming that we indeed get the expression before factorization.

Therefore the correct answer is answer A.