Example 14. Solve the equation

$$
3^{2 x^{2}+6 x-9}+4 \cdot 15^{x^{2}+3 x-5}=3 \cdot 5^{2 x^{2}+6 x-9}
$$
Solution. Using the properties of exponents, we can represent the given equation as

$$
\frac{1}{3} \cdot 9^{x^{2}+3 x-4}+\frac{4}{15} \cdot 15^{x^{2}+3 x-4}=\frac{3}{5} 25^{x^{2}+3 x-4}
$$

Divide both sides of equation (10) by $15^{x^{2}+3 x-4}$ and introduce the notation $y=(3 / 5)^{x^{2}-3 x-4}$. We obtain the equation

$$
y / 3+4 / 15=3 /(5 y)
$$

which has a unique positive root $y_{1}=1$.

Thus, equation (9) is equivalent to the equation

$$
(3 / 5)^{x^{2}+3 x-4}=1
$$

i.e.

$$
x^{2}+3 x-4=0
$$

from which it follows that the set of all solutions to the equation consists of two numbers: -4 and 1.