The fourth zero is a real number because complex numbers come in conjugate pairs. Having another complex zero means it will come with a pair, giving the function five zeros, which is not possible.
Work Step by Step
RECALL: The Conjugate Pairs Theorem states that if $a+bi$ is a zero of a polynomial function with real coefficients, then $a-bi$ is also a zero of the function. The function's degree is four so it has four zeros. Having $2, 1+2i,$ and $1-2i$ as zeros mean that the fourth zero must be a real number because complex zeros come in pairs.