Answer
Domain: (-∞, ∞)
Range: (-∞, -2)
Asymptote: y = -2
See graph below
Work Step by Step
This problem just asks for the graph of the function, domain, range, and asymptote.
Given $g(x) = -e^{x+1} - 2$
The graph in black represents $e^x$
$g(x)$ (in red) is a transformation of 1 unit to the left, reflection about the x-axis, and down 2 unit
Domain: (-∞, ∞)
Range: (-∞, -2)
Asymptote: y = -2