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> Find $m$ so that the roots of the equation $x^2 - (3m - 1)x + m^2 - 2 = 0$ lie inside the interval $(1, 5)$.
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Having $x\_1, x\_2 \in (1, 5)$, we get $x\_1 + x\_2 \in (2, 10)$ and $x\_1x\_2 \in (1, 25)$. By solving these two inequations I got $m \in \left( \sqrt3, \frac{11}{3} \right)$. However, this result is wrong. By substituting $m$ with $2$, for example, the given conditions are not reached.

So, please, explain what I did wrong and how to get the right solution.

Thank you in advance!