Perform the indicated operation and simplify. Assume that the variable represents a nonnegative real number \( (9-\sqrt{x+2})^{2} \)

To simplify the expression \( (9 - \sqrt{x+2})^{2} \), we can use the formula for the square of a binomial, which is \( (a - b)^{2} = a^{2} - 2ab + b^{2} \). Here, let \( a = 9 \) and \( b = \sqrt{x+2} \). Calculating each term: 1. \( a^{2} = 9^{2} = 81 \) 2. \( b^{2} = (\sqrt{x+2})^{2} = x + 2 \) 3. \( 2ab = 2(9)(\sqrt{x+2}) = 18\sqrt{x+2} \) Now, substituting these results back into the formula: \[ (9 - \sqrt{x+2})^{2} = 81 - 18\sqrt{x+2} + (x + 2) \] Combining like terms: \[ 81 + x + 2 - 18\sqrt{x+2} = x + 83 - 18\sqrt{x+2} \] Thus, the simplified expression is: \[ x + 83 - 18\sqrt{x+2} \]