Answer: \(m = \frac{5 + \sqrt{13}}{2} \text{ or } m = \frac{5 - \sqrt{13}}{2}\)
Half of \(-5\) is \(-\frac{5}{2}\); add \(\frac{25}{4}\) to both sides.
\[m^2 - 5m + \frac{25}{4} = -3 + \frac{25}{4}\]\[\left(m - \frac{5}{2}\right)^2 = \frac{13}{4}\]\[m - \frac{5}{2} = \pm\frac{\sqrt{13}}{2}\]\(m = \frac{5 + \sqrt{13}}{2}\) or \(m = \frac{5 - \sqrt{13}}{2}\).