= Solution
Use <integration by parts> with a <complex conjugate>, so that the argument also covers complex-valued <eigenfunctions>. The coefficient functions are understood to be real. For a nonzero <eigenfunction> $u$ with <Dirichlet boundary conditions>,
$$
\begin{aligned}
\lambda\int_\alpha^\beta |u|^2\,dx
&=\int_\alpha^\beta\overline u\,Mu\,dx\\
&=\bigl[a\overline u\,u'\bigr]_\alpha^\beta-\int_\alpha^\beta a|u'|^2\,dx+\int_\alpha^\beta b|u|^2\,dx\\
&=-\int_\alpha^\beta a|u'|^2\,dx+\int_\alpha^\beta b|u|^2\,dx.
\end{aligned}
$$
The denominator is strictly positive and the numerator is real. Consequently the <eigenvalue> satisfies
$$
\boxed{\lambda=\frac{\int_\alpha^\beta(b|u|^2-a|u'|^2)\,dx}{\int_\alpha^\beta|u|^2\,dx}\in\mathbb R.}
$$
This is the <Rayleigh quotient> for the sign convention used here. Positivity of $a$ is unnecessary for this particular reality argument; real coefficients and disappearance of the endpoint term are what matter.
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