Take it easy!

$\begin{aligned} \text{Since } x^2 - 15x + 1 & = 0 \\ \Rightarrow x^2 & = 15x - 1 \quad \quad \small \color{#3D99F6}{\text{Dividing throughout by }x} \\ x & = 15 - \frac{1}{x} \end{aligned}$

Since $a$ and $b$ are roots of the equation, then $a = 15 - \dfrac{1}{a}$ and $b = 15 - \dfrac{1}{b}$ , therefore,

$\begin{aligned} \left(\frac{1}{a} - 1\right)^{-2} + \left(\frac{1}{b} - 1\right)^{-2} & = (-a)^{-2} + (-b)^{-2} \\ & = \frac{1}{a^2} + \frac{1}{b^2} \\ & = \frac{a^2+b^2}{a^2b^2} \\ & = \frac{(a+b)^2-2ab}{(ab)^2} \quad \quad \small \color{#3D99F6}{\text{By Vieta's formulas: }a+b = 15, \space ab = 1} \\ & = \frac{15^2-2}{1} = \boxed{223} \end{aligned}$