Answer
X-intercept: None
Y-intercept: (0, -9/4)
Vertical asymptote: x = -2
Domain: (-∞, -2) U (-2, ∞)
Horizontal asymptote: y=-2
Range: (-∞, -2)
Graph is below
Work Step by Step
$r(x)=\frac{-2x^2 - 8x - 9}{x^2 + 4x + 4}$
-2x^2 - 8x - 9= 0
$2 (x^2 + 4x + 4) = -9 + 8$
$2 (x+2)^2 = -1$
No x-intercept
y-intercept is the ratio of the constants, which is -9/4
Thus, the y-intercept is at (0, -9/4)
Vertical asymptotes are when the denominator is equal to 0
$x^2 + 4x + 4 = 0$
$(x+2)^2 = 0$
x = -2
So, domain is from (-∞, -2) U (-2, ∞)
Horizontal asymptote is the ratio of the constants of the leading term (with equal degree)
-2/1 = -2
Thus, the horizontal asymptote is at y=-2
So the range is from (-∞, -2)