(a) The complex number z satisfies z² - (3 + i)z + m + 2i = 0 with m a real number, and the equation has a real root. Find m. (b) For real numbers a, b, c the polynomial P(z) = 2z⁴ + a z³ + b z² + c z + 3 has both 2 and i among the roots of P(z) = 0. Find a.

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(a) m = 2; (b) a = -11/2. (a) Splitting the equation into real and imaginary parts gives x = 2 and m = 2. (b) Since i and -i are roots and 2 is a root, the fourth root is found from the product of roots = 3/2, then a comes from the sum of roots.

(a) The complex number z satisfies z² - (3 + i)z + m + 2i = 0 with m a real number, and the equation has a real root. Find m. (b) For real numbers a, b, c the polynomial P(z) = 2z⁴ + a z³ + b z² + c z + 3 has both 2 and i among the roots of P(z) = 0. Find a.

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