Polynomial factor problem

$x^2 - 3x + 2 = (x-1)(x-2)$.

Since this is a factor of $f(x) = x^4 - px^2 + q$, $f(1) = 0$ and $f(2) = 0$.

1. $f(1) = 1 - p + q = 0 \implies p - q = 1$
2. $f(2) = 16 - 4p + q = 0 \implies 4p - q = 16$

Subtracting (1) from (2): $3p = 15 \implies p = 5$ $q = 5 - 1 = 4$

Thus, $(p,q) = (5,4)$.