= Solution
Blow up the distinct singular points of the <integral projective curve>, once at each original point, and let $C'$ be its <strict transform of an algebraic subvariety>. Write $H$ for the pullback of a <projective line> and $E_i$ for the <algebraic exceptional divisors>. The <multiplicity of a plane curve at a point> and the <intersection formula for blowing up a surface> give
$$
[C']=dH-\sum_i m_iE_i,\qquad
H^2=1,\qquad H\cdot E_i=0,\qquad E_i\cdot E_j=-\delta_{ij}.
$$
The <canonical divisor formula for a surface blowup> gives $K=-3H+\sum_iE_i$. Applying the <arithmetic adjunction formula on a smooth surface>, rather than a formula requiring a smooth strict transform, yields
$$
2p_a(C')-2=d^2-3d-\sum_i m_i(m_i-1),
$$
and hence
$$
p_a(C')=\frac{(d-1)(d-2)}2-\frac12\sum_i m_i(m_i-1).
$$
The <strict transform of an algebraic subvariety> is still an <integral projective curve>. Its <arithmetic genus> is $h^1(C',\mathcal O_{C'})\ge0$, even if some singularities remain. Therefore the <plane curve singularity multiplicity bound> is
$$
\boxed{\sum_i m_i(m_i-1)\le(d-1)(d-2).}
$$
The argument uses only one <blowup of a smooth algebraic surface> at each original singular point; it does not assume these blowups resolve every singularity.
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