I've been learning about the resolution of singularities for plane curves, and have become stuck at exercise 7.15 of Fulton's Algebraic Curves (page 91 of the PDF).
The question is:
Let $F=F_1, \ldots, F_m$ be a sequence of quadratic transformations of a curve $F$ such that $F_m$ has only ordinary multiple points. Let $P_{i1}, P_{i2}, \ldots$ be the points on $F_i$ introduced, as in (7)(c) on Page 90 of the PDF, in going from $F_{i-1}$ to $F_i.$ (These are called "neighboring singularities"). If $n = \text{deg}(F),$ show that
$$ (n-1)(n-2) \geq \sum_{P\in F} m_P(F) \left( m_P(F)-1 \right) + \sum_{ i>1, j} m_{P_{ij}} (F_i) \left( m_{P_{ij}} (F_i)-1\right). $$
Just to be clear, $m_P(F)$ denotes the multiplicity of the point $P$ on the curve with equation $F=0.$
I really don't know how to even approach this problem. It certainly looks related to Theorem 2 on page 60 of the PDF which shows
If $F$ is an irreducible curve of degree $n,$ then $$\sum_{P\in F} m_P(F) \left( m_P(F)-1 \right) \leq (n-1)(n-2)$$
But I can't see how to resolve the question.