6. Solve the equation $\sqrt{3 x^{2}-5 x-4}+\sqrt{3 x^{2}-5 x-8}=2$.
Solution: $\sqrt{3 x^{2}-5 x-4}+\sqrt{3 x^{2}-5 x-8}=2$, Domain of Definition (ODZ) $3 x^{2}-5 x-8 \geq 0$.
$t=3 x^{2}-5 x-4, \quad \sqrt{t}+\sqrt{t-4}=2, \quad t+t-4+2 \sqrt{t(t-4)}=4, \quad 2 \sqrt{t(t-4)}=8-2 t$, $\sqrt{t-(t-4)}=4-t, \quad(t \leq 4) . \quad t^{2}-4 t=16-8 t+t^{2}, \quad t=\frac{8}{2}=4 . \quad 3 x^{2}-5 x-4=4$, $3 x^{2}-5 x-8=0, x_{1,2}=\frac{5 \pm \sqrt{25+96}}{6}=\frac{5 \pm 11}{6}=\left[\begin{array}{l}\frac{8}{3} \\ -1\end{array}\right.$.

Verification: $\sqrt{3 \cdot \frac{8^{2}}{3^{2}}-\frac{40}{3}-4}+\sqrt{8-8}=2, \sqrt{3+5-4}+\sqrt{3+5-8}=2$.

Answer: $\frac{8}{3} ;-1$.