Domain of the function $f(x)=\log(\sin^{-1}\sqrt{x^2+3x+2})$ is:

$[\frac{-3-\sqrt{5}}{2},-2)∪(-1,\frac{-3+\sqrt{5}}{2}]$

Defined for $\sin^{-1}\sqrt{x^2+3x+2}>0⇒\sqrt{x^2+3x+2}>0$ true for all x.

Also, $-1≤\sqrt{x^2+3x+2}≤1$

Here $x^2+3x+2>0$ and $x^2+3x+2≤1;(x+1)(x+2)>0$ and $\frac{-3-\sqrt{5}}{2}≤x≤\frac{-3+\sqrt{5}}{2}$

$x∈(-∞,-2)∪(-1,∞)$ and $x∈[\frac{-3-\sqrt{5}}{2},\frac{-3+\sqrt{5}}{2}]⇒x∈[\frac{-3-\sqrt{5}}{2},-2)∪(-1,\frac{-3+\sqrt{5}}{2}]$