if 3 and -4 are the roots of the polynomial G(x) =x^4+3x^3+px^2-31x+2k, find the values of k,p, and the other root

Answer 1

a little synthetic division yields
G(-4) = 16p+k+188
G(3) = 9p+k+69
By the Remainder Theorem, both values must be zero, so we have p = -17 and q = 42
x^4+3x^3-17x^2-31x+84 = (x+4)(x-3)(x^2-2x+7)
Now just solve the quadratic as usual.