Let's analyze each of the given algebraic expressions to determine which one is a polynomial.
### Expression 1: \( 4x^2 - 3x + \frac{2}{x} \)
- The terms are \( 4x^2 \), \(-3x\), and \( \frac{2}{x} \).
- A polynomial must not have any variable terms in the denominator. The term \( \frac{2}{x} \) can be rewritten as \( 2x^{-1} \), which involves a negative exponent.
- Therefore, \( 4x^2 - 3x + \frac{2}{x} \) is not a polynomial.
### Expression 2: \( -6x^3 + x^2 - \sqrt{5} \)
- The terms are \(-6x^3\), \( x^2 \), and \(-\sqrt{5} \).
- A polynomial can include variables raised to non-negative integer powers and constants.
- The term \( \sqrt{5} \) is simply a constant, which is allowed in polynomials. The other terms, \(-6x^3\) and \( x^2 \), have non-negative integer exponents.
- Therefore, \( -6x^3 + x^2 - \sqrt{5} \) is a polynomial.
### Expression 3: \( 8x^2 + \sqrt{x} \)
- The terms are \( 8x^2 \) and \( \sqrt{x} \).
- The term \( \sqrt{x} \) can be rewritten as \( x^{1/2} \), which involves a fractional exponent.
- Polynomials cannot have fractional exponents.
- Therefore, \( 8x^2 + \sqrt{x} \) is not a polynomial.
### Expression 4: \( -2x^4 + \frac{3}{2x} \)
- The terms are \(-2x^4\) and \( \frac{3}{2x} \).
- The term \( \frac{3}{2x} \) can be rewritten as \( \frac{3}{2} x^{-1} \), which involves a negative exponent.
- Polynomials cannot have terms with negative exponents.
- Therefore, \( -2x^4 + \frac{3}{2x} \) is not a polynomial.
Given the analysis, the only expression that meets the criteria to be considered a polynomial is:
[tex]\[
-6x^3 + x^2 - \sqrt{5}
\][/tex]
Thus, the expression that is a polynomial is:
[tex]\[
-6x^3 + x^2 - \sqrt{5}
\][/tex]
