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

Factorize the given expression 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, in reverse order:

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

To understand that:

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

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: $a^3$,

Continue to extract the largest common factor for the numbers 2,4,8, which is clearly the number 2 given that 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:

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

After extracting the common factor outside of 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?" Fill in the missing parts inside of the parentheses whilst making sure that the sign of the term that 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 obtain the expression before factorization.

Therefore the correct answer is answer A.