Okay, let's tackle this question. The patient is a 45-year-old woman with a hard, mobile breast lump. 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. The key features that raise suspicion for malignancy include being hard, fixed (not mobile), irregular in shape, and associated with other signs like skin dimpling or nipple discharge. However, this lump is described as hard but mobile. That's interesting because mobility can sometimes be a sign of a benign lesion, like a fibroadenoma. But age is a factor here—45 is a time when breast cancer risk increases.

The next investigation would typically be imaging. Mammography is standard, especially in women over 35-40. But maybe an ultrasound is preferred first in certain cases. Wait, the options might include both. Let me think: in younger women, ultrasound is often first because their breast tissue is denser. But at 45, mammography might be more appropriate. Alternatively, the next step after a clinical exam could be a mammogram. If the options include mammography, that's likely the answer. But if the question is about imaging, maybe ultrasound is first. Hmm. Also, a core biopsy might be the next step if imaging is inconclusive. But the question is about the next investigation after a physical exam. So probably imaging.

Wait, the options here are not provided, but the correct answer is to be selected. The user hasn't given the options, but the correct answer is probably mammography or ultrasound. Let me check standard guidelines. For a palpable breast mass, the initial investigations are usually imaging. In a woman in her 40s, mammography is typically the first step, but ultrasound can be used as a complementary tool. If the lump is solid and hard, a biopsy might be needed, but that's after imaging. So maybe mammography is the next step. Alternatively, if the lump is mobile, it's more likely benign, but still needs imaging to confirm. The next investigation would be either mammography or ultrasound. The answer here is likely mammography. Let me confirm. The American College of Radiology guidelines suggest that for a palpable lump in a woman over 35, mammography is the initial test. So the correct answer is mammography. Let's structure the explanation accordingly.

**Core Concept**
Breast lumps in women aged 40+ require imaging to differentiate benign from malignant pathology. A hard, mobile lump may suggest fibroadenoma (benign) or atypical features, but imaging is essential for definitive evaluation. The standard initial investigation is **mammography** to assess density, calcifications, and architectural distortion.

**Why the Correct Answer is Right**
Mammography is the gold standard for evaluating breast abnormalities in women ≥35 years. It detects microcalcifications (often associated with malignancy) and provides structural information about the lump. A hard, mobile lesion may still harbor malignant features (e.g., spiculated margins) visible on imaging. Ultrasound is typically used first in younger women with dense breasts but is secondary in this age group.

**Why Each Wrong Option is Incorrect**
**Option A:** *Clinical examination alone is insufficient for definitive diagnosis.*
**Option B:** *Core

✓ Correct Answer: C. Mammography