Correct Option: C) 32

Step-by-step Solution:1. Simplify the Expression:The given expression is \(P = \frac{x^4+16x^2+64}{x^2}\).We can divide each term in the numerator by \(x^2\):\(P = \frac{x^4}{x^2} + \frac{16x^2}{x^2} + \frac{64}{x^2}\)\(P = x^2 + 16 + \frac{64}{x^2}\)2. Introduce a Substitution:Let \(t = x^2\). Since \(x\) is a non-zero real number, \(x^2\) must be positive. Therefore, \(t > 0\).Substituting \(t\) into the simplified expression, we get:\(P = t + 16 + \frac{64}{t}\)3. Apply AM-GM Inequality:The Arithmetic Mean - Geometric Mean (AM-GM) inequality states that for any non-negative real numbers \(a\) and \(b\), \(\frac{a+b}{2} \ge \sqrt{ab}\), which implies \(a+b \ge 2\sqrt{ab}\). Equality holds when \(a=b\).We apply this to the positive terms \(t\) and \(\frac{64}{t}\):\(t + \frac{64}{t} \ge 2\sqrt{t \cdot \frac{64}{t}}\)\(t + \frac{64}{t} \ge 2\sqrt{64}\)\(t + \frac{64}{t} \ge 2 \cdot 8\)\(t + \frac{64}{t} \ge 16\)4. Determine the Minimum Value of P:From the AM-GM inequality, the minimum value of \(t + \frac{64}{t}\) is 16.Substituting this back into the expression for \(P\):\(P \ge 16 + 16\)\(P \ge 32\)Thus, the minimum value of the expression is 32.5. Condition for Equality:The equality in AM-GM holds when the terms are equal, i.e., \(t = \frac{64}{t}\).\(t^2 = 64\)Since \(t > 0\), we take the positive root, \(t = 8\).As \(t = x^2\), we have \(x^2 = 8\), which means \(x = \pm\sqrt{8} = \pm 2\sqrt{2}\). These are non-zero real values, confirming that the minimum is achievable.The final answer is 32.