Question: Factor completely. Select “Prime” if the polynomial cannot be factored. \[ x^2 y^2 + 12x^2 y +…

Factor completely. Select “Prime” if the polynomial cannot be factored.

\[ x^2 y^2 + 12x^2 y + 27x^2 \]

\[ x^2 y^2 + 12x^2 y + 27x^2 = \boxed{\phantom{0}} \]

Prime

Solution

Factor the given polynomial completely: The polynomial is: \[ x^2y^2 + 12xy + 27x^2 \] First, check if there is a common factor in each term. The terms are: - \(x^2y^2\) - \(12xy\) - \(27x^2\) The greatest common factor of all terms is \(x\). Factor out \(x\) from each term: \[ x(xy^2 + 12y + 27x) \] Now, examine the expression \(xy^2 + 12y + 27x\). This expression does not have a straightforward factorization, and no further common factor exists. Thus, the original polynomial cannot be factored further. Therefore, the polynomial is prime.