Final answer:
To factor the polynomial 48x² + 48xy² + 12y⁴ completely, we first look for common factors. Then, we factor out the common factors to obtain the completely factored form 12y²(4x² + 4xy² + y²).
Step-by-step explanation:
To factor the polynomial 48x2 + 48xy2 + 12y4 completely, we look for common factors first. In this case, we can factor 12 out of each term:
48x2 + 48xy2 + 12y4 = 12(4x2 + 4xy2 + y4)
Now, we can see that there is a common factor of y2 in each term, so we can factor it out:
12(4x2 + 4xy2 + y4) = 12y2(4x2 + 4xy2 + y2)
Therefore, the polynomial is completely factored as 12y2(4x2 + 4xy2 + y2).