Find all rational zeros of f. Then use the depressed equation to find all roots of the equation f(x)=0.
** i got the answer that 6 is the rational zero but not sure with second part of problem
With x = 6 as one solution we must have:
(x-6)(x2 + ax + b) = x3 - 37x + 6
or x3 + ax2 + bx - 6x2 - 6ax - 6b = x3 - 37x + 6
or x3 + (a-6)x2 + (b-6a)x - 6b = x3 - 37x + 6
So we must have a = 6 and b = -1 for the LHS to equal the RHS.
The depressed equation is thus the quadratic: x2 + 6x - 1 = 0
Can you take it from here?