Let x be the length of a diagonal of a face of a cube and y be the length of a diagonal of the cube. If x + y = (5 + 2√6) units, then what is the total surface area of the cube?

If side of cube = a, then face diagonal x = a√2 and space diagonal y = a√3. So xy = a√2 · a√3 = a²√6. Then 6xy = 6a²√6. The total surface area of a cube = 6a². We need 6a² to equal one of the expressions. With x + y = a(√2 + √3) = 5 + 2√6, we need a² explicit. Actually, xy = a²√6 doesn't directly give 6a². Re-examining: total surface area = 6a². We have x = a√2, so x² = 2a², giving a² = x²/2. So 6a² = 3x². Alternatively, x·y = a²√6 gives a² = xy/√6, so 6a² = 6xy/√6 = √6·xy. Given x + y = 5 + 2√6, solve: a(√2 + √3) = 5 + 2√6. Squaring: a²(5 + 2√6) = (5 + 2√6)², so a² = 5 + 2√6. Then 6a² = 30 + 12√6 = 6(5 + 2√6) = 6(x + y). So total surface area = 6(x + y), option (a).