Rational function with given vertical and oblique asymptotes, a local maximum, a modulus equation and the reciprocal of a translated graph

The function \(f\) is defined by \(f(x)=\dfrac{x^2+bx+c}{x+d}\) for \(x\in\mathbb{R}\), \(x\ne-d\), where \(b\), \(c\) and \(d\) are integers.

The line \(x=-2\) is a vertical asymptote and the line \(y=x-3\) is an oblique asymptote of the graph of \(y=f(x)\).

1. Write down \(d\).[1]
2. Show that \(b=-1\).[2]
3. The function \(f\) has a local maximum where \(x=-5\).
   Find \(c\).[4]
4. Sketch the curve \(y=f(x)\) for \(-20\leq x\leq20\), showing clearly any points where it meets the axes, the coordinates of its local maximum and local minimum points, and the equations of its asymptotes.[4]
5. The equation \(\lvert f(x)\rvert=k\), where \(k\in\mathbb{R}\), has exactly four distinct real roots.
   Find all possible values of \(k\).[2]
6. The curve \(y=g(x)\) is the image of the curve \(y=f(x)\) under a translation by the vector \(\begin{pmatrix}3\\ -5\end{pmatrix}\).
   Find the equation of each vertical asymptote of the graph of \(y=\dfrac{1}{g(x)}\).[4]