\( \begin{aligned} \text { 5. } \frac{m^{2}+7 m}{3 m} \div \frac{49-m^{2}}{3 m-21} & = \\ & = \\ & =\end{aligned} \)

To solve the problem, start by simplifying the expression step by step. First, let's rewrite the equation: \[ \frac{m^{2}+7 m}{3 m} \div \frac{49-m^{2}}{3 m-21} \] Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{m^{2}+7 m}{3 m} \times \frac{3 m - 21}{49 - m^{2}} \] Now, factor the parts of the fractions. The numerator of the first fraction can be factored as: \[ m(m + 7) \] The denominator \(3m - 21\) can be factored as: \[ 3(m - 7) \] And notice that \(49 - m^{2}\) can be factored as: \[ (7 - m)(7 + m) \] Putting it all together: \[ \frac{m(m + 7)}{3m} \times \frac{3(m - 7)}{(7 - m)(7 + m)} \] Cancelling \(3m\) in the first part gives us: \[ \frac{m + 7}{3} \times \frac{3(m - 7)}{(7 - m)(7 + m)} \] Now, notice that \(7 - m\) can be factored as \(-1(m - 7)\): \[ \frac{m + 7}{3} \times \frac{3(-1)(m - 7)}{-(m + 7)(m - 7)} \] Next, the \(3\) cancels and we can simplify further. The final simplified expression becomes: \[ \frac{1(m - 7)}{1} = m - 7 \] So the answer to the expression is: \[ \boxed{m - 7} \]