(ctgx+3)^2+2a^3+3a^2=(a^2+2a+3)*(ctgx+3) equation

Given the equation
$$2 a^{3} + 3 a^{2} + \left(\cot{\left(x \right)} + 3\right)^{2} = \left(\cot{\left(x \right)} + 3\right) \left(a^{2} + 2 a + 3\right)$$
transform
$$2 a^{3} + 3 a^{2} + \left(\cot{\left(x \right)} + 3\right)^{2} - \left(\cot{\left(x \right)} + 3\right) \left(a^{2} + 2 a + 3\right) = 0$$
$$- \left(\cot{\left(x \right)} + 3\right) \left(a^{2} + 2 a + 3\right) + \left(2 a^{3} + 3 a^{2} + \left(\cot{\left(x \right)} + 3\right)^{2}\right) = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Expand the expression in the equation
$$2 a^{3} + 3 a^{2} + \left(w + 3\right)^{2} - \left(w + 3\right) \left(a^{2} + 2 a + 3\right) = 0$$
We get the quadratic equation
$$2 a^{3} - a^{2} w - 2 a w + w^{2} - 6 a + 3 w = 0$$
This equation is of the form
$$a\ w^2 + b\ w + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = - a^{2} - 2 a + 3$$
$$c = 2 a^{3} - 6 a$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(- a^{2} - 2 a + 3\right)^{2} - 1 \cdot 4 \cdot \left(2 a^{3} - 6 a\right) = - 8 a^{3} + \left(- a^{2} - 2 a + 3\right)^{2} + 24 a$$
The equation has two roots.
$$w_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$w_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$w_{1} = \frac{a^{2}}{2} + a + \frac{\sqrt{- 8 a^{3} + \left(- a^{2} - 2 a + 3\right)^{2} + 24 a}}{2} - \frac{3}{2}$$
Simplify
$$w_{2} = \frac{a^{2}}{2} + a - \frac{\sqrt{- 8 a^{3} + \left(- a^{2} - 2 a + 3\right)^{2} + 24 a}}{2} - \frac{3}{2}$$
Simplify
do backward replacement
$$\cot{\left(x \right)} = w$$
$$\cot{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acot}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{acot}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acot}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acot}{\left(\frac{a^{2}}{2} + a + \frac{\sqrt{- 8 a^{3} + \left(- a^{2} - 2 a + 3\right)^{2} + 24 a}}{2} - \frac{3}{2} \right)}$$
$$x_{1} = \pi n + \operatorname{acot}{\left(\frac{a^{2}}{2} + a + \frac{\sqrt{- 8 a^{3} + \left(- a^{2} - 2 a + 3\right)^{2} + 24 a}}{2} - \frac{3}{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acot}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acot}{\left(\frac{a^{2}}{2} + a - \frac{\sqrt{- 8 a^{3} + \left(- a^{2} - 2 a + 3\right)^{2} + 24 a}}{2} - \frac{3}{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acot}{\left(\frac{a^{2}}{2} + a - \frac{\sqrt{- 8 a^{3} + \left(- a^{2} - 2 a + 3\right)^{2} + 24 a}}{2} - \frac{3}{2} \right)}$$