First we multiply the expression by 1:
[x + Sqrt(x^2 + 3x)]*(x - Sqrt(x^2 + 3x)]/[x - Sqrt(x^2 + 3x)] =
[x^2 - x^2 - 3x]/[x - Sqrt(x^2 + 3x)] = -3x/[x - Sqrt(x^2 + 3x)]
Now we're going to multiply the above by 1 = (1/-x)/(1/-x)...why -1/x?
Because we're going to "put" 1/-x inside the square root in the denominator, but as x --> -oo we HAVE to put inside the root something positive! Said this, pay atention to the fact that 1/-x gets into the root
as 1/x^2 , so :
-3x/[x - Sqrt(x^2 + 3x)]*(1/-x)/(1/-x) = 3/[-1 - Sqrt(1 + 3/x)] --> 3/-2
by arithmetic of limits and since 3/x --> 0 when x --> +/- oo.
You can now grab your personal calculator and input low values for x into the function, say x = -1,000 or x = -100,000, and convince yourself the limit is -3/2 indeed.