For these types of problems, its best to translate your given information into mathematical expressions/equations. Given:
P(x) = ax^4 + bx^3 + cx^2 + dx + 4 with a,b,c,d integer values we also have
P(x) = (x - SQRT(2))*(fx^3 + gx^2 + hx + k)
= (x - SQRT(2)(x - r)(jx^2 + ox + p)
where SQRT(2) + r is rational and > 0.
You can also use other results like the rational root theorem for your problem.
What course is this for by the way?