Answer
$x=183+ 8\sqrt{ 517}\approx 364.90 $
Work Step by Step
Given $$\sqrt{6x-5}=4+\sqrt{5x+2}.$$ First we square both sides to eliminate one of the radicals: $$\begin{aligned}
(\sqrt{6x-5})^2&=(4+\sqrt{5x+2})^2\\
6x-5&=16+8\sqrt{5x+2}+5x+2\\
x-23&=8\sqrt{5x+2}.
\end{aligned}$$ We square both sides again to eliminate the other radical: $$\begin{aligned}
(x-23)^2&=(8\sqrt{5x+2})^2\\
x^2-46x+529&=64(5x+2)\\
x^2-46x+529&=320x+128\\
x^2-366x+401&=0.
\end{aligned}$$ Solve the equation: $$\begin{aligned}
x&=\frac{-(-366)\pm\sqrt{(-366)^2-4(1)(401)}}{2(1)}\\
&=\frac{366\pm 16\sqrt{517}}{2}\\
&=183\pm8\sqrt{517}.
\end{aligned}$$ The solutions are: $$
\begin{aligned}
x&=183-8\sqrt{517}\\
x&=183+8\sqrt{517}.
\end{aligned}$$ Check: $$
\begin{aligned}
\sqrt{6(183-8\sqrt{517})-5}&\stackrel{?}{=}4+\sqrt{5(183-8\sqrt{517})+2}\\
\sqrt{1093-48\sqrt{517}}&\stackrel{?}{=}4+\sqrt{917-40\sqrt{517}}\\
24-\sqrt{517}&\stackrel{?}{=}4+\sqrt{517}-20\\
1.26&\not=6.74\\\\
\sqrt{6(183+8\sqrt{517})-5}&\stackrel{?}{=}4+\sqrt{5(183+8\sqrt{517})+2}\\
\sqrt{1093+48\sqrt{517}}&\stackrel{?}{=}4+\sqrt{917+40\sqrt{517}}\\
24+\sqrt{517}&\stackrel{?}{=}4+\sqrt{517}+20\\
24+\sqrt{517}&=24+\sqrt{517}\checkmark.
\end{aligned}$$ The only solution is $$x=183+8\sqrt{517}\approx 364.90.$$
The solution is $x=183+ 8\sqrt{ 517}\approx 364.90 $
