Answer
$$\frac{1}{2}a^3b^3-\frac{1}{3}a^2b^5+\frac{5}{9}ab^6.
$$
Work Step by Step
We multiply each term of the polynomial by the monomial and then add the products as follows
$$
\frac{2}{3} a b^{3}\left(\frac{3}{4} a^{2}-\frac{1}{2} a b^{2}+\frac{5}{6} b^{3}\right)=\frac{1}{2}a^3b^3-\frac{1}{3}a^2b^5+\frac{5}{9}ab^6.
$$
