Question: Factor completely. Select “Prime” if the polynomial cannot be factored. \[ 14r^2 - 15rs - 9s^2…

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

\[ 14r^2 - 15rs - 9s^2 \]

\[ 14r^2 - 15rs - 9s^2 = \]

Solution

The given polynomial is \(14r^2 - 15rs - 9s^2\). To factor this, we can use the method of grouping or trial and error for simple quadratics. The polynomial is in the form \(ax^2 + bx + c\) where: - \(a = 14\), - \(b = -15\), and - \(c = -9\). First, we look for two numbers that multiply to \(ac = 14 \times (-9) = -126\) and add up to \(b = -15\). The numbers are \(6\) and \(-21\). Rewrite the middle term using these numbers: \[ 14r^2 + 6rs - 21rs - 9s^2 \] Group the terms: \[ (14r^2 + 6rs) + (-21rs - 9s^2) \] Factor out the greatest common factor from each group: \[ 2r(7r + 3s) - 3s(7r + 3s) \] Notice that \(7r + 3s\) is common in both terms: \[ (7r + 3s)(2r - 3s) \] Thus, the factored form of the polynomial is \[ (7r + 3s)(2r - 3s) \]