If $x$ is an odd integer, then which of the following is true?

### Concept & Logic The properties of odd and even numbers dictate the parity of algebraic expressions. When determining parity, substituting a base odd integer simplifies the algebraic rules. ### Step-by-Step Solution * **Given:** $x$ is an odd integer. * **Deduction:** Let's assume a simple odd integer for $x$, such as $x = 1$. * **Calculation:** * Test Option (a): $5(1) - 2 = 3$ (This is odd, so the statement is false). * Test Option (b): $5(1)^2 + 2 = 5(1) + 2 = 7$ (This is odd, so the statement is true). * Test Option (c): $5(1)^2 + 3 = 5(1) + 3 = 8$ (This is even, so the statement is false). ### Exam Strategy & Shortcut **Value Substitution:** Never try to memorize complex parity rules (like $Odd \times Odd + Even = Odd$) under time pressure. Immediately substitute $x = 1$ (for odd) or $x = 2$ (for even) to instantly verify the correct option. ### Common Pitfall Students sometimes pick an overly large or negative odd number (like $-3$ or $7$), which introduces unnecessary arithmetic complexity and increases the chance of minor calculation errors. Stick to $1$. ### Final Answer Therefore, the correct answer is **$5x^2 + 2$ is odd**.