Calculating slant asymptotes of radical function

Calculating slant asymptotes of radical function

I'm trying to calculate the slant asymptotes of the function
$\sqrt{x^2+2x+2}$. I've found out that the gradients of the asymptotes are
1 for $x\rightarrow+\infty$ and -1 for $x\rightarrow-\infty$. I've also
found out that the constant of the positive asymptote is 1. Intuitively, I
know the constant of the negative asymptote is -1, but I'm struggling to
show it through calculation. I need to evaluate this to find it:
$$\lim_{x\rightarrow-\infty} \sqrt{x^2+2x+2} + x $$
without using l'Hôpital's rule (for the purposes of the assignment I'm not
supposed to know how to use it.) I have tried rationalizing the numerator
using the conjugate but I just end up with an undefined value.