Solve the following equation:

$$
(2 x+3)^{2}+(3 x-5)^{2}-\sqrt{13 x^{2}-18 x+20}=44
$$
The equation can be transformed into the following form:

$$
13 x^{2}-18 x-10-\sqrt{13 x^{2}-18 x+20}=0
$$

Let

$$
\sqrt{13 x^{2}-19 x+20}=y
$$

then:

$$
y^{2}-y-30=0
$$

from which

$$
y_{1}=6, y_{2}=-5
$$

thus

(1 $\left.^{\circ}\right) 13 x^{2}-18 x+20=36$

(2 $\left.2^{\circ}\right) 13 x^{2}-18 x+20=25$.

From these equations, we obtain:

$$
x_{1}=2, x_{2}=-\frac{8}{13}, x_{3}=\frac{9+\sqrt{146}}{13}, x_{4}=\frac{9-\sqrt{146}}{13} .
$$

(Denes König, Budapest.)

Number of solutions: 52.