For what value of the prime p, does the polynomial have a root of multiplicity >1 over the field of

characteristic p..

p=2, 3, 5 or 7.

Please help.

The answer is p = 3.

Take for example the Galois field GF(3) = {0, 1, 2} with Char(GF(3)) = 3.
Since in GF(3) 6 = 3 = 0 and -1 = 2, we have
x⁴ + x + 6 = x⁴ + x = x(x³ + 1) = x(x³ + 3x² + 3x + 1) = x(x + 1)³ = x(x - 2)³, i.e. 2 is a root of multiplicity 3 (and of course 0 is a simple root).

Over GF(2) x⁴ + x + 6 = x⁴ + x = x(x³ + 1) = x(x + 1)(x² + x + 1) and the roots are 0 and 1 (both simple). The 3rd factor has no roots in GF(2).
Over GF(5) x⁴ + x + 6 = x⁴ + x + 1 = (x + 2)(x³ + 3x² + 4x + 3) - simple root -2 = 3.
Finally over GF(7) x⁴ + x + 6 = (x + 3)(x³ + 4x² + 2x + 2) - simple root -3 = 4.