Correct Option: BStep 1: Simplify the expressionThe given expression is \$\\frac{x^2 - 4x + 7}{x - 2}\$.We can perform polynomial long division or algebraic manipulation to simplify the expression:\$\\frac{x^2 - 4x + 7}{x - 2} = \\frac{x(x-2) - 2x + 7}{x - 2}\$\$= \\frac{x(x-2) - 2(x-2) - 4 + 7}{x - 2}\$\$= \\frac{x(x-2)}{x - 2} - \\frac{2(x-2)}{x - 2} + \\frac{3}{x - 2}\$\$= x - 2 + \\frac{3}{x - 2}\$Step 2: Apply AM-GM inequalityLet \$y = x - 2\$.Since we are given that \$x > 2\$, it implies that \$x - 2 > 0\$, so \$y > 0\$.The expression can now be written as \$y + \\frac{3}{y}\$.For any two positive real numbers, the Arithmetic Mean (AM) is always greater than or equal to the Geometric Mean (GM). That is, for positive a and b, \$\\frac{a+b}{2} \\ge \\sqrt{ab}\$.Applying this to \$y\$ and \$\\frac{3}{y}\$:\$\\frac{y + \\frac{3}{y}}{2} \\ge \\sqrt{y \\cdot \\frac{3}{y}}\$\$\\frac{y + \\frac{3}{y}}{2} \\ge \\sqrt{3}\$\$y + \\frac{3}{y} \\ge 2\\sqrt{3}\$Step 3: Determine when equality holdsThe equality in the AM-GM inequality holds when the two numbers are equal, i.e., \$y = \\frac{3}{y}\$.\$y^2 = 3\$.Since \$y > 0\$, we take the positive square root: \$y = \\sqrt{3}\$.Substituting back \$y = x - 2\$, we get \$x - 2 = \\sqrt{3}\$, which means \$x = 2 + \\sqrt{3}\$.Since \$2 + \\sqrt{3} \\approx 2 + 1.732 = 3.732\$, which is indeed greater than 2, this value of x is valid and the minimum is attainable.Step 4: Conclusion and analysis of optionsThe minimum value of the expression is \$2\\sqrt{3}\$.Option A (\$\\sqrt{3}\$) is less than the derived minimum value, hence it is not achievable. Option C (\$\$2 + \\sqrt{3}\$) represents the value of x at which the minimum is attained, not the minimum value of the expression itself. Option D (4) is greater than \$2\\sqrt{3}\$ (approximately 3.464) and therefore not the minimum.Therefore, the correct option is B.