This answer is to explain my comment more thoroughly.
You are looking for a polynomial $r(x) = ax + b$ such that
$$x^{73}-2x^{15}+3x-1 = q(x)(x-1)^2+r(x)$$
for some polynomial $q(x)$. If we replace $y = x - 1$, i.e. $x = y + 1$, we get an equivalent problem:
$$(y+1)^{73}-2(y+1)^{15}+3(y+1)-1 = q(y+1)y^2+r(y+1).$$
The $q(y+1)y^2$ contains no constant or $y$ terms, and $r(y + 1) = ay + a + b$ contains no $y^2$ terms or higher. So, we simply need to find the constant and $y$ terms in order to figure out $r$.
We can do this with binomial theorem. In particular, we know
$$(y + 1)^n = \sum_{k=0}^n \binom{n}{k} y^k = \sum_{k=0}^n \frac{n!}{k!(n-k)!} y^k.$$
So, the constant terms and $y$ terms can be extracted, simply by taking $k = 0$ and $k = 1$ respectively. We have $\binom{n}{0} = 1$ and $\binom{n}{1} = n$. Ignoring quadratic and higher order terms, the left hand side becomes:
$$(1 + 73y + \ldots) - 2(1 + 15y + \ldots) + 3(1 + y) - 1 = 1 + 46y + \ldots$$
Therefore, $r(y + 1) = ay + a + b = 1 + 46y$, which implies $a = 46$, and $b = -45$, i.e. $r(x) = 46x - 45$.