Pattern-recognition first: on renal immunofluorescence there are three deposit patterns. Linear means anti-GBM (Goodpasture), granular means immune-complex disease (lupus, post-infectious), and pauci-immune means ANCA vasculitis. The biopsy here shows smooth linear IgG, so the diagnosis is anti-GBM disease.
The clinical story fits: a nephritic syndrome with hematuria, edema and a creatinine of $9$ mg% reflecting aggressive glomerular damage. To confirm the cause, the single most useful test is the serum assay for $anti$-$GBM$ antibodies, which directly identifies the autoantibody driving the disease.
Eliminating the rest: ANA chases lupus, urine immunoelectrophoresis chases myeloma light chains, and HIV RNA chases HIV nephropathy, but none of these explain a linear IgG biopsy. The answer is option (b), anti-GBM antibodies.