If \( \int \frac{(x-1) d x}{x^{2} \sqrt{2 x^{2}-2 x+1}}=\frac{\sqrt{f(x)}}{g(x)}+C \), where \( ...

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If \( \int \frac{(x-1) d x}{x^{2} \sqrt{2 x^{2}-2 x+1}}=\frac{\sqrt{f(x)}}{g(x)}+C \), where \( f(x) \) is a quadratic expression and \( g(x) \) is a monic linear expression.
(A) \( f(x)=2 x^{2}-2 x+1 \)
(B) \( g(x)=x+1 \)
(C) \( g(x)=x \)
(D) \( f(x)=2 x^{2}-2 x \)
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