1. The problem involves simplifying the expression given and solving the equation: $$x^2 + \frac{x}{2} = \left(x + \frac{x}{1}\right)^2 - 2 = 16^2 - 2 = 256 - 2 = 254$$ 2. Interpreting the expression for clarity, we assume: $$\left(x + \frac{x}{1}\right)^2 - 2 = 16^2 - 2$$ 3. Simplify inside the parentheses: $$x + \frac{x}{1} = x + x = 2x$$ 4. So, the expression becomes: $$ (2x)^2 - 2 = 256 - 2 = 254 $$ 5. Simplify the left side: $$ 4x^2 - 2 = 254 $$ 6. Add 2 to both sides: $$ 4x^2 = 256 $$ 7. Divide both sides by 4: $$ x^2 = 64 $$ 8. Take the square root of both sides: $$ x = \pm 8 $$ 9. Final answer: $$ \boxed{x = 8 \text{ or } x = -8} $$ This step-by-step shows how the expression is simplified and solved for $x$.