To rationalize the denominator of the expression \(\frac{7}{5 + \sqrt{31}}\), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \(5 - \sqrt{31}\).
The expression becomes:
\[ \frac{7(5 - \sqrt{31})}{(5 + \sqrt{31})(5 - \sqrt{31})} \]
Now, let's simplify the denominator:
\[ (5 + \sqrt{31})(5 - \sqrt{31}) = 5^2 - (\sqrt{31})^2 = 25 - 31 = -6 \]
Now, simplifying the numerator:
\[ 7(5 - \sqrt{31}) = 35 - 7\sqrt{31} \]
Putting it all together, we have:
\[ \frac{35 - 7\sqrt{31}}{-6} \]
This simplifies to:
\[ -\frac{35 - 7\sqrt{31}}{6} \]
We can separate the terms in the numerator:
\[ -\frac{35}{6} + \frac{7\sqrt{31}}{6} \]
So the final simplified answer is:
\[ -\frac{35}{6} + \frac{7\sqrt{31}}{6} \]
Thus, the answer is:
\[ \frac{7\sqrt{31} - 35}{6} \]