Example 3 Given that $\alpha, \beta$ are the roots of the equation $x^{2}-7 x+8=0$, and $\alpha>\beta$. Without solving the equation, find the value of $\frac{2}{\alpha}+3 \beta^{2}$.
Solution: By Vieta's formulas, we know $\alpha+\beta=7, \alpha \beta=8$. Let $M=\frac{2}{\alpha}+3 \beta^{2}, N=\frac{2}{\beta}+3 \alpha^{2}$, it is easy to find that $M+N=100 \frac{3}{4}$,
$M-N=-\frac{85 \sqrt{17}}{4}$.
(1) + (2) gives
$$
M=\frac{2}{\alpha}+3 \beta^{2}=\frac{1}{8}(403-85 \sqrt{17}) \text {. }
$$