Okay, let's tackle this question. The patient is a 45-year-old woman with a hard, mobile lump in the breast. The question is asking for the next investigation. First, I need to recall the standard approach to breast lumps.

Breast lumps can be benign or malignant. Hard and mobile might suggest a benign lesion like a fibroadenoma, but malignancy isn't ruled out. The next step usually involves imaging. Mammography is a common first-line investigation, especially for women over 40. However, in some cases, an ultrasound might be preferred first, especially if the patient is under 35 or if the lump is palpable but not obvious on mammogram. But since this patient is 45, mammography is more likely the next step. Alternatively, a core needle biopsy could be considered if imaging is inconclusive. But the question is about the next investigation, so probably imaging first.

Wait, the options aren't provided, but the correct answer is supposed to be one of the standard investigations. Let me think again. The core concept here is the evaluation of a breast mass. The standard initial investigation is usually mammography followed by ultrasound. If there's a palpable lump, ultrasound can help characterize it as solid or cystic. However, for a 45-year-old, mammography is typically the first step. If the lump is mobile and hard, it might be a fibroadenoma, but malignancy is less likely if it's mobile. Still, imaging is necessary. The next investigation would be mammography. If the mammogram is inconclusive, then ultrasound or biopsy. So the correct answer is likely a mammogram.

Now, the wrong options: maybe options like fine needle aspiration, core biopsy, or MRI. FNA is less commonly used now, and core biopsy is more definitive but might come after imaging. MRI is not first-line. So the correct answer is mammography.

**Core Concept**
Breast lump evaluation requires imaging to distinguish benign from malignant pathology. Mammography is the primary investigation for women over 40 due to its sensitivity in detecting microcalcifications and architectural distortions characteristic of malignancy.

**Why the Correct Answer is Right**
The correct investigation is **mammography**. It uses X-rays to visualize breast tissue, identifying subtle abnormalities like microcalcifications (common in ductal carcinoma in situ) or spiculated masses (suggestive of malignancy). For a 45-year-old with a hard, mobile lump, mammography is first-line to assess for cancer, even if fibroadenomas (benign, mobile lumps) are more likely in younger women.

**Why Each Wrong Option is Incorrect**
**Option A:** *Fine-needle aspiration (FNA)* is less specific and not first-line; it risks sampling error and cannot distinguish benign from malignant lesions reliably.
**Option B:** *Ultrasound* is useful for characterizing solid vs. cystic masses but is secondary to mammography in older women.
**Option D:** *MRI* is reserved for high-risk patients (e.g., BRCA carriers) or staging, not initial evaluation.

**Clinical Pearl**
Never skip mammography for a palpable breast lump in women over 40—it remains the gold standard for early cancer detection. "Mobile" does not rule out malignancy; 1

✓ Correct Answer: C. Mammography