Task B-2.1. Solve the equation

$$
3^{1+4 x+2 x^{2}}+2^{1+4 x+2 x^{2}}=5 \cdot 6^{x(x+2)}
$$
Solution. Let's rewrite the equation in another form:

$$
3^{4 x+2 x^{2}} \cdot 3+2^{4 x+2 x^{2}} \cdot 2=5 \cdot 2^{x^{2}+2 x} \cdot 3^{x^{2}+2 x}
$$

Divide the equation by $3^{2 x^{2}+4 x}$ and introduce the substitution $\left(\frac{2}{3}\right)^{x^{2}+2 x}=t$.

The given equation transforms into the quadratic equation $2 t^{2}-5 t+3=0$.

Its solutions are $t_{1}=1, t_{2}=\frac{3}{2}$.

The solutions to the original equation are $x_{1}=0, x_{2}=-2, x_{3}=-1$.