4. Given equation A: $x^{2}+p_{1} x+q_{1}=0$, equation B: $x^{2}+p_{2} x+q_{2}=0$, where $p_{1} 、 p_{2} 、 q_{1} 、 q_{2}$ are all real numbers, and satisfy $p_{1} p_{2}=2\left(q_{1}+q_{2}\right)$. Then $(\quad)$.
(A) Both A and B must have real roots
(B) Neither A nor B has real roots
(C) At least one of A and B has real roots
(D) Whether A and B have real roots cannot be determined
4. (C).
$$
\Delta_{1}=p_{1}^{2}-4 q_{1}, \Delta_{2}=p_{2}^{2}-4 q_{2} \text {. }
$$

If equations 甲 and 乙 have no real roots, then
$$
\Delta_{1}<0, \Delta_{2}<0, \Delta_{1}+\Delta_{2}<0 \text {, }
$$

which means
$$
\begin{array}{l}
p_{1}^{2}+p_{2}^{2}-4\left(q_{1}+q_{2}\right)<0, \\
p_{1}^{2}+p_{2}^{2}-2 p_{1} p_{2}<0, \\
\left(p_{1}-t_{2}\right)^{2}<0 .
\end{array}
$$

This is impossible. Therefore, at least one of the two equations has real roots.