Which of the following expression is not a polynomial?

Options

A. `(1)/(x+1)`
B. `x^(2)+1`
C. `4m^(2)+9m+1`
D. `5y^(2)-6`.

Explanation

To determine which of the given expressions is not a polynomial, we first need to understand the definition of a polynomial. A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents.

Let's analyze each of the given expressions step by step:

1. **Expression 1: \( \frac{1}{x} + 1 \)**
- This expression can be rewritten as \( x^{-1} + 1 \).
- The term \( x^{-1} \) has a negative exponent, which violates the definition of a polynomial (since the exponent must be a non-negative integer).
- Therefore, this expression is **not a polynomial**.

2. **Expression 2: \( b(x^2 + 1) \)**
- This expression consists of \( b \) multiplied by \( x^2 + 1 \).
- The term \( x^2 \) has a non-negative integer exponent (2), and the constant term (1) is also acceptable.
- Thus, this expression is a polynomial.

3. **Expression 3: \( 4m^2 + 9m + 1 \)**
- Here, we have three terms: \( 4m^2 \), \( 9m \), and \( 1 \).
- All the exponents (2 for \( m^2 \), 1 for \( m \), and 0 for the constant term) are non-negative integers.
- Therefore, this expression is a polynomial.

4. **Expression 4: \( 5y^2 - 6 \)**
- This expression consists of \( 5y^2 \) and a constant term (-6).
- The exponent of \( y^2 \) is 2 (a non-negative integer), and the constant term is acceptable.
- Hence, this expression is also a polynomial.

After analyzing all the expressions, we conclude that the first expression \( \frac{1}{x} + 1 \) is not a polynomial.

### Final Answer:
The expression that is not a polynomial is \( \frac{1}{x} + 1 \).