To simplify the expression \( \frac{5x}{x+9} - x + \frac{36}{x+9} \), let's follow these steps:

### Step 1: Combine the fractions
The common denominator of the two fractions is \(x+9\). We can rewrite the expression as:

\[
\frac{5x}{x+9} + \frac{36}{x+9} - x
\]

Now, combine the two fractions:

\[
\frac{5x + 36}{x+9} - x
\]

### Step 2: Simplify the expression
We now have:

\[
\frac{5x + 36}{x+9} - x
\]

We can write the expression as:

\[
\frac{5x + 36}{x+9} - \frac{x(x+9)}{x+9}
\]

This gives us a common denominator \(x+9\):

\[
\frac{5x + 36 - x(x+9)}{x+9}
\]

### Step 3: Expand and simplify the numerator
Expand \(x(x+9)\):

\[
x(x+9) = x^2 + 9x
\]

Now subtract it from \(5x + 36\):

\[
5x + 36 - (x^2 + 9x) = -x^2 - 4x + 36
\]

### Step 4: Write the final simplified expression
The expression simplifies to:

\[
\frac{-x^2 - 4x + 36}{x+9}
\]

This is the simplified form of the expression.

### Do you have any questions, or would you like more details on any part of the process?

Here are some related questions you might find interesting:
1. How can I factor the quadratic expression in the numerator?
2. Can this expression have any restrictions on the domain?
3. What happens if I substitute \(x = -9\) into the original expression?
4. How can I further simplify rational expressions involving more complex denominators?
5. Can this expression be simplified using polynomial division?

**Tip:** Always check the denominator for any restrictions to ensure the expression is valid!

Learn how to simplify the rational expression 5x/(x+9) - x + 36/(x+9) using a step-by-step method. This solution covers concepts such as finding a common denominator, expanding binomials, and simplifying quadratic expressions. Ideal for students in Grades 9-10.

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