Answer: First, we need to find the composite function (fg)(x):
(fg)(x) = f(g(x))
= f(-4x-3)
= (-4x-3)^2 + (-4x-3) + 9 (substituting g(x) into f(x))
= 16x^2 + 24x + 18
Therefore, (fg)(x) = 16x^2 + 24x + 18.
Next, we need to find the quotient function (f/g)(x):
(f/g)(x) = f(x) / g(x)
= (x^2 + x + 9) / (-4x - 3) (substituting f(x) and g(x))
To simplify this expression, we can use polynomial long division or synthetic division. Using synthetic division, we get:
-4 | 1 1 9
|_____-4__ 12
| 1 -3 21
Therefore, (f/g)(x) = -4x + 3 - 21 / (-4x - 3)
Simplifying further, we get:
(f/g)(x) = -4x + 3 + (21/4)(1/(x + 3/4))
Therefore, (f/g)(x) = -4x + 3 + (21/4)(1/(x + 3/4)).
Explanation: