Step 1 :Given a polynomial of degree 10, with leading coefficient 1, and roots at x=0, x=-3, and x=-11 with multiplicities 3, 4, and 3 respectively.
Step 2 :The polynomial can be written in the form: \(P(x) = a*(x - r1)^{m1} * (x - r2)^{m2} * ... * (x - rn)^{mn}\)
Step 3 :Substituting the given values into the formula, we get the polynomial: \(P(x) = x^{10} + 45x^{9} + 813x^{8} + 7577x^{7} + 39219x^{6} + 113751x^{5} + 173151x^{4} + 107811x^{3}\)
Step 4 :\(\boxed{P(x) = x^{10} + 45x^{9} + 813x^{8} + 7577x^{7} + 39219x^{6} + 113751x^{5} + 173151x^{4} + 107811x^{3}}\)