The curve C has an equation $$y = \frac{ax^2+bx+c}{x+d},$$ where $a$, $b$, $c$, and $d$ are constants. The curve cuts the $y$-axis at $(0,-2)$ and has asymptotes $x=2$ and $y = x + 1$.
From a previous part of the question, I have obtained $d = -2$. I am now asked to obtain the values of $a$, $b$ and $c$. Here is what I have done,
First, I would find $c$ since I know the curve cuts the $y$ axis at $(0,-2)$ $$\begin{align} \frac{a(0)^2+b(0)+c}{0-2} & = -2 \\ \frac{c}{-2} & = -2 \\ c & = 4 \end{align}$$
I do not know what to do next to find $a$ and $b$ but I tried to substitute $y = x+1$ into the equation of the curve and compared the coefficients but that did not work. Could someone please guide me how to do so? I find questions like these unusual since my textbook only showed how to find the asymptotes and graph such functions.
Hint:
$$ \lim_{x \to \infty}\dfrac{ax^2+bx+4}{x(x-2)}=1 $$ and $$ \lim_{x \to \infty}\left[\dfrac{ax^2+bx+4}{x-2}-(x+1) \right]=0 $$