5.63 Given that $m, n, p, q$ are real numbers. For the quadratic trinomials
$$x^{2}+m x+n \text { and } x^{2}+p x+q$$

to have real roots, and for the roots of one polynomial to be separated by the roots of the other polynomial
[Solution] Let $x_{1}, x_{2}$ be the roots of the quadratic trinomial $x^{2} + m x + n$, and $x_{3}, x_{4}$ be the roots of the quadratic trinomial $x^{2} + p x + q$.

If these two pairs of roots are mutually separated, that is, $x_{1}, x_{2}, x_{3}, x_{4}$ are real numbers, and one of $x_{3}, x_{4}$ is within the interval $\left(x_{1}, x_{2}\right)$, while the other is outside this interval.

Thus, there are two possible scenarios: either the differences $x_{3} - x_{1}, x_{3} - x_{2}$ have the same sign, and the differences $x_{4} - x_{1}, x_{4} - x_{2}$ have opposite signs; or the differences $x_{3} - x_{1}, x_{3} - x_{2}$ have opposite signs, and the differences $x_{4} - x_{1}, x_{4} - x_{2}$ have the same sign. In both cases, we have
$$\left(x_{3} - x_{1}\right)\left(x_{3} - x_{2}\right)\left(x_{4} - x_{1}\right)\left(x_{4} - x_{2}\right) < 0 .$$

Since $x_{1} + x_{2} = -m, x_{1} x_{2} = n$, we have
$$\begin{aligned}
\left(x_{3} - x_{1}\right)\left(x_{3} - x_{2}\right) & = x_{3}^{2} - \left(x_{1} + x_{2}\right) x_{3} + x_{1} x_{2} \\
& = x_{3}^{2} + m x_{3} + n, \\
\left(x_{4} - x_{1}\right)\left(x_{4} - x_{2}\right) & = x_{4}^{2} - \left(x_{1} + x_{2}\right) x_{4} + x_{1} x_{2} \\
& = x_{4}^{2} + m x_{4} + n .
\end{aligned}$$

Therefore, inequality (1) can be rewritten as
$$\left(x_{3}^{2} + m x_{3} + n\right)\left(x_{4}^{2} + m x_{4} + n\right) < 0 .$$

Expanding the left side of the inequality, noting that
$$\begin{array}{l}
x_{3} + x_{4} = -p, x_{3} x_{4} = q \text{ and } \\
x_{3}^{2} + x_{4}^{2} = \left(x_{3} + x_{4}\right)^{2} - 2 x_{3} x_{4} = p^{2} - 2 q,
\end{array}$$

we have
$$\begin{aligned}
& \left(x_{3}^{2} + m x_{3} + n\right)\left(x_{4}^{2} + m x_{4} + n\right) \\
= & x_{3}^{2} x_{4}^{2} + m x_{3} x_{4}\left(x_{3} + x_{4}\right) + m^{2} x_{3} x_{4} + n\left(x_{3}^{2} + x_{4}^{2}\right) + m n\left(x_{3} + x_{4}\right) + n^{2} \\
= & q^{2} - m p q + m^{2} q + n\left(p^{2} - 2 q\right) - m n p + n^{2} \\
= & (n - q)^{2} + m q(m - p) + n p(p - m) \\
= & (n - q)^{2} + (m - p)(m q - n p) .
\end{aligned}$$

Therefore, inequality (2) can be written as
$$(n - q)^{2} + (m - p)(m q - n p) < 0$$

This means that if the roots of the quadratic trinomials $x^{2} + m x + n$ and $x^{2} + p x + q$ are mutually separated, then the coefficients of these quadratic trinomials satisfy condition (3).