Yep you do polynomial division, I'm not sure how you were taught. I was taught to try and find a factor.

f(5) = 125-75+10-60 = 0.

Therefore (n-5) is a factor and we can divide through using long division and solve the remaining quadratic.

or compare coefficients: (n-5)(an^2+bn+c) = n^3-3n^2+2n-60

For example compare coefficients of x^0 (the constant): -5c = -60 therefore c=12