Okay, let's tackle this question. The patient is a 67-year-old male with symptoms like headache, recurrent infections, multiple punched-out lytic lesions on the skull X-ray, and lumbago (lower back pain). The question is asking which investigation would help establish the diagnosis.

First, I need to recall the conditions that present with these symptoms. Multiple punched-out lytic lesions are classic for multiple myeloma. Other possibilities might include metastatic cancer, but the combination with lumbago (which could be due to spinal involvement) and recurrent infections (due to impaired immunity) points more towards multiple myeloma.

The core concept here is the diagnostic approach to multiple myeloma. The key investigations for multiple myeloma include serum and urine protein electrophoresis to detect monoclonal proteins (M proteins), serum free light chains, bone marrow biopsy for plasma cell infiltration, and imaging studies like X-rays or MRI for bone lesions.

The question asks for the investigation that helps establish the diagnosis. The options are not given, but common tests for this scenario would be serum protein electrophoresis (SPEP), urine protein electrophoresis (UPEP), bone marrow biopsy, and possibly serum calcium or creatinine levels. Another possibility is a complete blood count (CBC) to check for anemia, which is common in myeloma.

Wait, but the key diagnostic test for multiple myeloma is detecting the presence of M proteins. SPEP can detect IgG or IgA, while UPEP detects Bence Jones proteins. Bone marrow biopsy confirms plasma cell infiltration. However, the initial step is often SPEP and UPEP. If those are positive, further tests confirm. But the question is about the investigation that helps establish the diagnosis. So the correct answer is likely serum protein electrophoresis.

Looking at the options, if one of them is SPEP, that's the answer. If not, maybe bone marrow biopsy. But in the absence of options, I need to think. The question mentions multiple lytic lesions, which are a hallmark of myeloma. The main diagnostic tests are SPEP, UPEP, and bone marrow. The most direct test to confirm the presence of M proteins is SPEP. So the correct investigation would be serum protein electrophoresis.

Now, the incorrect options might include things like a simple CBC, which can show anemia but isn't diagnostic. Or maybe a plain X-ray, which is already mentioned in the question. Alternatively, a test like serum calcium might be elevated in myeloma, but that's a lab value, not an investigation. So the wrong options would be other tests that don't confirm the diagnosis directly.

Clinical pearl: Remember that multiple myeloma is diagnosed by a combination of clinical features and lab tests, with SPEP and UPEP being the key initial tests. Also, bone marrow biopsy is used to confirm plasma cell infiltration if needed.

**Core Concept**
The clinical presentation of multiple lytic bone lesions, recurrent infections, and lumbago is classic for multiple myeloma. Diagnosis hinges on detecting monoclonal gammopathy via serum protein electrophoresis (SPEP) and urine protein electrophoresis (UPEP), along with bone marrow assessment.

**Why the Correct Answer is Right**
Serum protein electrophoresis (SPEP) is

✓ Correct Answer: A. Protein electrophoresis