Question: Factor the trinomial. \[ x^2y^2 + 19xy + 45 \] Select the correct choice below and, if…

Factor the trinomial.

\[ x^2y^2 + 19xy + 45 \]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. \( x^2y^2 + 19xy + 45 = \boxed{\phantom{0}} \) (Factor completely.)

B. The polynomial is prime.

Solution

To factor the trinomial \( \frac{x^2}{y^2} + 19xy + 45 \): First, we try to express the trinomial in a form that allows for factoring. We need to find two numbers that multiply to the constant term and add up to the linear coefficient. Here, the constant term is \( 45 \) and the linear coefficient is \( 19 \). The numbers that multiply to \( 45 \) and add up to \( 19 \) are \( 4.5 \) and \( 10 \), but these aren’t correct. Let’s find factors that work: Possible pairs of factors of 45: - \(1\) and \(45\) - \(3\) and \(15\) - \(5\) and \(9\) Among these, \(4\) and \(15\) do not have a sum of \(19\). Next, try \(5\) and \(9\): - They multiply to \(45\) - They add to \(14\), which is incorrect for the linear coefficient. However, upon further inspection, it seems like a re-evaluation of integers was necessary. After re-evaluation: The trinomial is not factorable using integer coefficients for the quadratic form given in the question. So, we choose: Option B: The polynomial is prime.