<s>[INST] Finding all real roots of the equation $(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$ 
Find all real roots of the equation
$$(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$$

I tried squaring the equation, but the degree of the equation became too high and unmanageable. I also tried substitutions, but it didn't work out correctly. This question was in my weekly class worksheet as were this and this question which I previously asked.
Any help will be appreciated.
Thanks.
 [/INST] Let $\sqrt{x+2}=a$ and $\sqrt{x+7}= b$.
So, $x + 1 = a^2 -1$
$x+6 = b^2 -1$
$ x^2 + 7x + 12= (x+3)(x+4)= (b^2-4)(a^2+2)$
Substituting and solving the system of equations.
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