Explanation

This uses the denesting identity (p + √q)² = (p² + q) + 2p√q. Writing 7 + 4√3 as (p + √q)² and matching coefficients, (2 + √3)² = 4 + 3 + 4√3 = 7 + 4√3, so a = 2 + √3 (the positive root). Rationalising, 1/a = 1/(2 + √3) = 2 − √3. Adding: a + 1/a = (2 + √3) + (2 − √3) = 4. Hence option (c). The distractors 2 and 3 come from misidentifying the denested form (e.g. using 1 + √6 instead of 2 + √3), and 7 from simply reading off the constant term.