How do you develop a recurrence relation for the function $f(n) = 5n^2 +3$, where $n \in \mathbb{Z}^+$? On an exam of mine, I was asked to find a  recurrence relation for the function $f(n) = 5n^2 +3$, where $n \in \mathbb{Z}^+$. I needed to provide a base case and the actual relation itself. I know the base case is for $n = 1$, were $f(1) = 8$, but I have no idea how to derive the relation from here.
The professor's answer key is as follows, but I don't understand where the intuition/motivation comes from for this solution:
$f(1) = 8, f(n) = 5n^2 + 3 = 5(n - 1)^2 + 3 + 5(2n - 1) = f(n - 1) + 10n - 5$
Where do I start? The above steps seem, at least to me, to be arbitrarily and magically plucked from nowhere...

The motivation behind rewriting $f(n)$ this way, is to acquire an equation of the form $$f(n)=g(f(n-1)),$$
i.e. we have to rewrite $f(n)$, so that we get some function of $f(n-1)$.
Here, this function is
$$g(x)=x+10n-5.$$
Now you can directly deduct the recurrence relation, by plugging $g$ into $x_{n+1}=g(x_n)$.
Here, we get the result
$$x_1=8,~~x_{n+1}=x_n+10n-5$$
