1. **State the problem:** We need to determine which expressions always yield an even value when $x$ is an odd number. 2. **Recall important rules:** - An odd number is any integer of the form $2k+1$ where $k$ is an integer. - Even numbers are multiples of 2. - Multiplying an even number by any integer results in an even number. - Adding two odd numbers results in an even number. 3. **Analyze each expression:** **Expression 1: $4x$** - Since $4$ is even, $4x$ is $4 \times x$. - Multiplying an even number by any integer results in an even number. - Therefore, $4x$ is always even regardless of $x$. **Expression 2: $2x + 3$** - $2x$ is even because 2 is even. - $3$ is odd. - Even + Odd = Odd. - So $2x + 3$ is odd when $x$ is odd. **Expression 3: $x^2$** - The square of an odd number is odd. - Because $(2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$ which is odd. - So $x^2$ is odd when $x$ is odd. **Expression 4: $5 + x$** - $5$ is odd. - $x$ is odd. - Odd + Odd = Even. - So $5 + x$ is even when $x$ is odd. 4. **Final answer:** The expressions that always have an even value when $x$ is odd are: - $4x$ - $5 + x$