1. **Stating the problem:** Calculate the product of the binomials $$\left(\frac{3}{5} ab^{2} + 6\right)\left(\frac{3}{5} ab^{2} - 6\right)$$ using algebraic identities. 2. **Formula used:** This is a difference of squares pattern: $$ (A + B)(A - B) = A^{2} - B^{2} $$ where $$ A = \frac{3}{5} ab^{2} $$ and $$ B = 6 $$. 3. **Apply the formula:** $$\left(\frac{3}{5} ab^{2}\right)^{2} - 6^{2}$$ 4. **Calculate each square:** $$\left(\frac{3}{5}\right)^{2} a^{2} b^{4} - 36$$ 5. **Simplify the fraction squared:** $$\frac{9}{25} a^{2} b^{4} - 36$$ 6. **Final answer:** $$\boxed{\frac{9}{25} a^{2} b^{4} - 36}$$ This is the simplified product of the given binomials.