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The given expression to simplify is:

\[
\frac{a^2 + 9}{3a + a}
\]

Let's proceed with the simplification step by step.

1. **Simplify the Denominator**: Combine like terms in the denominator:
   \[
   3a + a = 4a
   \]

   So, the expression becomes:
   \[
   \frac{a^2 + 9}{4a}
   \]

2. **Simplify the Numerator**: In this case, the numerator \( a^2 + 9 \) does not factor further using real numbers (since \( a^2 + 9 \) is not a difference of squares, but a sum of squares). Thus, no further factorization is possible using real numbers.

Since there is no common factor in both the numerator and the denominator, this is the simplified form of the given expression:

\[
\frac{a^2 + 9}{4a}
\]

Would you like more details or have any questions? Here are some related questions that might help deepen your understanding:

1. How can we simplify a fraction with a polynomial numerator and denominator?
2. What are some common factoring techniques used to simplify expressions?
3. What happens when the numerator and denominator have a common factor?
4. How can you factor a sum of squares?
5. How do we handle expressions with more than one variable?

**Tip:** Always look for common factors in both the numerator and the denominator to simplify rational expressions.

Learn how to simplify the rational expression (a^2 + 9) / (4a) step-by-step. This problem involves rational expressions and polynomial simplification techniques. Suitable for students in Grades 9-12.

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