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The given expression is:

\[
\frac{x^4 + 9x^3}{2x^2}
\]

Let's simplify it step by step:

### Step 1: Factor the numerator if possible.
The numerator is \(x^4 + 9x^3\). We can factor out the common term \(x^3\):

\[
x^4 + 9x^3 = x^3(x + 9)
\]

So the expression becomes:

\[
\frac{x^3(x + 9)}{2x^2}
\]

### Step 2: Simplify the fraction.
Now, cancel out the common terms from the numerator and denominator. Since \(x^3\) is in the numerator and \(x^2\) is in the denominator, we can reduce \(x^3\) by dividing both the numerator and denominator by \(x^2\):

\[
\frac{x^3(x + 9)}{2x^2} = \frac{x(x + 9)}{2}
\]

### Final simplified expression:
\[
\frac{x(x + 9)}{2}
\]

This is the fully simplified form.

Would you like further clarification or details on any step? Here are some related questions:

1. How do you factor polynomials in general?
2. Can this expression be further simplified for specific values of \(x\)?
3. How does the degree of a polynomial affect simplification?
4. What is the difference between factoring and simplifying expressions?
5. Can you apply this method to other rational expressions?

**Tip:** Always check for common factors in both the numerator and denominator to simplify expressions more easily.

Simplify the expression completely if possible: (x^4 + 9x^3) / (2x^2).

Learn how to simplify the rational expression (x^4 + 9x^3) / (2x^2) with a detailed, step-by-step solution. This problem covers key algebraic concepts, including factoring polynomials and canceling common terms. Suitable for high school students in Grades 9-12.

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