Complex numbers differing by just the sign of the imaginary part are said to be conjugate.
So for example 2 + 3i and 2 - 3i are conjugate, -5 - 7i and -5 + 7i are conjugate, and in general
a + bi and a - bi where a and b are real numbers are conjugate.
They have the property that their product is always a real number, so for the general case,
(a + bi)(a - bi) = a^2 - abi + abi - (b^2)(i^2) = a^2 + b^2,
and as a particular example,
(2 + 3i)(2 - 3i) = 2^2 + 3^2 = 4 + 9 = 13.
For a polynomial equation, with real coefficients, complex roots always appear as conjugate pairs.
So, providing that the given equation does have real coefficients, it must have 3 - 4i as a second root.
The question doesn't actually state that P(x) has real coefficients, so it might have 3 - 4i as a second root, but then again it might not.
