If 3+√10 is a root then so is 3-√10. If those are roots, then [x-(3+√10)] and [x-(3-√10)] are factors. These can be written as [(x-3)-√10][(x-3)+√10). If you multiply those factors together and simplify you will get a difference of squares which simplifies to x2 -6x -1 . Use long polynomial division into the original function to get a quotient that is quadratic in x2. If the long polynomial division has a remainder you made a mistake.
Another option is to try synthetic division with 3+√10 followed by 3-√10, but that will get really messy.
The quotient of the long division should be 5x4 - 38x2 -16. Visualize letting U = x2 and solve 5U2 - 38U - 16 = 0. If you replace U with x2 and solve the resulting equations you will have all 6 real and complex roots.