Assume initially that x+3 is a factor of the numerator polynomial. Use zero -3 to divide by using synthetic division:
-3 | 5..16 -1 -10..6
......5 -15 -3..12 -6
......5....1 -4....2 | 0
Yes, it is a factor so the quotient is: 5x^3+x^2-4x+2. Note that this is true only if x not equal to -3, because this would create division by zero, which is disallowed.