algebra precalculus - Looking for a trick to solve $2sqrt {2x}+sqrt {2x+3}=sqrt {3x+2}+sqrt {6x+20}$

Consider the equation:

$$2\sqrt {2x}+\sqrt {2x+3}=\sqrt {3x+2}+\sqrt {6x+20}.$$

Find a trick ( if exists ) which allows to solve it elegantly i.e. with avoiding the systematic squaring.

(The systematic squaring inevitably leads to a fourth-degree equation:

$$\begin{align} 0 &= 207x^4-12564x^3+27738x^2+231084x-40401\\[6pt] &=9\left( 23x^2-1258x-4489\right) \left( x^2-6x+1\right)\;, \end{align}$$
so the answer is
$$x=\dfrac {629+\sqrt {498888}} {23}.$$