**Let \(a\) and \(b\) be the roots of the quadratic \(2x^2 - 8x + 7 = x^2 + 15x + 23.\) Compute \(a^4 + b^4\).**

We can first calculate the roots of the quadratic.

They are:

\(\frac{23+\sqrt{593}}{2}\) and \(\frac{23-\sqrt{593}}{2}\).

Now we can calculate the \(a^4 + b^4\) part.

\((\frac{23+\sqrt{593}}{2})^4+(\frac{23-\sqrt{593}}{2})^4\)

\(=\frac{314209+12903\sqrt{593}}{2}+\frac{314209-12903\sqrt{593}}{2}\)

\(=\boxed{314209}\)

Our final answer is \(\boxed{314209}.\)

*There is probably a much faster and more efficient way to solve this that someone more educated than I am will be able to teach you clearly