If alpha, beta are roots of x^2 - 3x + 1 = 0, find alpha^5 + beta^5.

If the roots of x^2 - 2px + p^2 + p - 3 = 0 are real and less than 3, find the largest integer value of p.

If x^2 - ax + b = 0 and x^2 - bx + a = 0 (a != b) have a common root, and a + b = 4, find a*b.

How many integral values of k in [-10, 10] make x^2 + kx + (k+3) = 0 have both roots as integers?

If alpha, beta are roots of x^2 - 4x + 1 = 0, find the value of alpha^4 + beta^4.