Solving $8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$ 
Solve the equation
  $$8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$$

I have this idea: set $$\sqrt{x+2}=a , x+2=a^2 , \sqrt{x-1}=b.$$
 So $$x-1=b^2 , 2a^2+6b^2 =8b-4$$ and $$x^2+x-2 =a^2b^2$$ and then I'd simplify, but it's still very hard to solve. 
Any hint is appreciated.

The domain gives $x\geq1$ and we need to solve that
$$\sqrt{x+2}-\sqrt{x-1}=7\sqrt{x^2+x-2}-8x+3.$$
Now, since $\sqrt{x+2}-\sqrt{x-1}\geq0,$ we obtain
$$7\sqrt{x^2+x-2}-8x+3\geq0$$ or
$$\frac{97-\sqrt{2989}}{30}\leq x\leq\frac{97+\sqrt{2989}}{30}.$$
Thus, we need to solve
$$\left(\sqrt{x+2}-\sqrt{x-1}\right)^2=\left(7\sqrt{x^2+x-2}-(8x-3)\right)^2$$ or
$$(112x-44)\sqrt{x^2+x-2}=113x^2-x-90$$ or
$$(112x-44)^2(x^2+x-2)=(113x^2-x-90)^2$$ or
$$225x^4-2914x^3+12669x^2-21468x+11972=0$$ or
$$(25x-146)(x-2)(9x^2-46x+41)=0,$$ which gives the answer:
$$\left\{2,\frac{23+4\sqrt{10}}{9}\right\}.$$
Also, your idea helps.
Indeed, we got the following system:
$$a^2-b^2=3$$ and
$$8(a^2-2)-3+a-b=7ab.$$
From the second equation we obtain:
$$b=\frac{8a^2+a-19}{7a+1},$$ which gives
$$a^2-\left(\frac{8a^2+a-19}{7a+1}\right)^2=3$$ or
$$(5a+14)(a-2)(3a^2-2a-13)=0$$ and the rest is smooth.
