1. The problem asks us to identify which expression is a polynomial from the given options. 2. A polynomial is an expression consisting of variables raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication only. No negative or fractional exponents are allowed. 3. Let's analyze each option: - A. $x^{-1} + 2x$: The term $x^{-1}$ has a negative exponent, so this is not a polynomial. - B. $\sqrt{x} + 1$: The square root of $x$ is $x^{\frac{1}{2}}$, which is a fractional exponent, so this is not a polynomial. - C. $x^{2} + 3x + 2$: All exponents are non-negative integers (2, 1, 0), so this is a polynomial. - D. $\frac{1}{x} + x^{2}$: The term $\frac{1}{x}$ is $x^{-1}$, which has a negative exponent, so this is not a polynomial. 4. Therefore, the only polynomial expression is option C: $x^{2} + 3x + 2$. Final answer: C. $x^{2} + 3x + 2$