Let z is complex number satisfying the equation, $z^2-(3+i)z+m+2i=0$, where m $\in$ R. Suppose the equation has a real root, then find the value of m.

Let the real root of the quadratic equation be z = k. Substituting z = k into z^2 - (3+i)z + m + 2i = 0 gives k^2 - 3k - ki + m + 2i = 0. Equating the real and imaginary parts to zero separately yields the system k^2 - 3k + m = 0 and -k + 2 = 0. Solving the imaginary equation gives k = 2, and substituting this into the real part gives 4 - 6 + m = 0, so m = 2.

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