Explanation

Factor x² − 4x + 3 = (x − 1)(x − 3). Since this divides f(x) = x⁴ + px² + q, both f(1) = 0 and f(3) = 0. This gives 1 + p + q = 0 and 81 + 9p + q = 0. Subtracting the first equation from the second gives 80 + 8p = 0, so p = −10, and then q = 9. Therefore p − q = −10 − 9 = −19, option A.