A polynomial is an expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a non-negative integral power.

Here the given polynomial is a trinomial.

To divide a polynomial by monomial, divide each term of the polynomial by the monomial.

Divide the trinomial by the monomial \(9a^{2}b^{2}\).

Simplify the terms which are under division.

Calculation:

Consider the polynomial \(\frac{-27a^{3}b^{4}-36a^{2}b^{3}+72a^{2}b^{5}}{9a^{2}b^{2}}\).

Divide each term of the polynomial by the monomial \(9a^{2}b^{2}\).

\(\frac{-27a^{3}b^{4}-36a^{2}b^{3}+72a^{2}b^{5}}{9a^{2}b^{2}} = (-27a^{3}\frac{b^{4}}{9}a^{2}b^{2})+(-36a^{2}\frac{b^{3}}{9}a^{2}b^{2})+(72a^{2}\frac{b^{5}}{9}a^{2}b^{2})=\)

\(= 3ab^{2}-4b+8b^{3}\)

The simplified value of the polynomial is \(3ab^{2}-4b+8b^{3}\).

Final statement:

The simplified value of the polynomial after division is equals to \(3ab^{2}-4b+8b^{3}\).

Here the given polynomial is a trinomial.

To divide a polynomial by monomial, divide each term of the polynomial by the monomial.

Divide the trinomial by the monomial \(9a^{2}b^{2}\).

Simplify the terms which are under division.

Calculation:

Consider the polynomial \(\frac{-27a^{3}b^{4}-36a^{2}b^{3}+72a^{2}b^{5}}{9a^{2}b^{2}}\).

Divide each term of the polynomial by the monomial \(9a^{2}b^{2}\).

\(\frac{-27a^{3}b^{4}-36a^{2}b^{3}+72a^{2}b^{5}}{9a^{2}b^{2}} = (-27a^{3}\frac{b^{4}}{9}a^{2}b^{2})+(-36a^{2}\frac{b^{3}}{9}a^{2}b^{2})+(72a^{2}\frac{b^{5}}{9}a^{2}b^{2})=\)

\(= 3ab^{2}-4b+8b^{3}\)

The simplified value of the polynomial is \(3ab^{2}-4b+8b^{3}\).

Final statement:

The simplified value of the polynomial after division is equals to \(3ab^{2}-4b+8b^{3}\).