Answer
$-1$
Work Step by Step
Recall, in order to solve problems involving order of operations, we use the PEMDAS rule.
First Priority: P - parentheses and other grouping symbols (including fraction bars)
Second Priority: E - exponents
Third Priority: M/D - Multiplication or division, whichever comes first from the left to the right
Fourth Priority: A/S - Addition or subtraction, whichever comes first from the left to the right
We follow order of operations to obtain that the expression, $
\dfrac{2-3[(4-5)^2+1]}{(3-1)^2}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{2-3[(-1)^2+1]}{(2)^2}
\\\\=
\dfrac{2-3[(-1)(-1)+1]}{(2)(2)}
\\\\=
\dfrac{2-3[1+1]}{4}
\\\\=
\dfrac{2-3[2]}{4}
\\\\=
\dfrac{2-6}{4}
\\\\=
\dfrac{-4}{4}
\\\\=
-1
.\end{array}
