https://file.aiquickdraw.com/mathbot/file/1727312487348u7birej3.jpg

The expression in the image is:

\[
\frac{3}{x} - \frac{6}{x + 3} - \frac{x}{(x + 3)^2}
\]

To simplify this expression, we need to combine the terms by finding a common denominator. Let's go step by step.

### Step 1: Identify a common denominator
The denominators are \( x \), \( x+3 \), and \( (x+3)^2 \). The least common denominator (LCD) for these is \( x(x + 3)^2 \).

### Step 2: Rewrite each term with the LCD as the denominator
We'll rewrite each fraction to have the common denominator \( x(x + 3)^2 \).

- For \( \frac{3}{x} \), multiply both the numerator and denominator by \( (x + 3)^2 \):
  \[
  \frac{3}{x} = \frac{3(x + 3)^2}{x(x + 3)^2}
  \]
  
- For \( \frac{6}{x+3} \), multiply both the numerator and denominator by \( x(x + 3) \):
  \[
  \frac{6}{x + 3} = \frac{6x(x + 3)}{x(x + 3)^2}
  \]

- For \( \frac{x}{(x+3)^2} \), the denominator is already \( (x+3)^2 \), so just multiply by \( x \) to get the LCD:
  \[
  \frac{x}{(x+3)^2} = \frac{x^2}{x(x + 3)^2}
  \]

### Step 3: Combine the terms
Now, we combine all three terms over the common denominator \( x(x + 3)^2 \):

\[
\frac{3(x + 3)^2}{x(x + 3)^2} - \frac{6x(x + 3)}{x(x + 3)^2} - \frac{x^2}{x(x + 3)^2}
\]

Simplify the numerators:

- Expand \( 3(x + 3)^2 \):
  \[
  3(x + 3)^2 = 3(x^2 + 6x + 9) = 3x^2 + 18x + 27
  \]

- Expand \( 6x(x + 3) \):
  \[
  6x(x + 3) = 6x^2 + 18x
  \]

So the expression becomes:

\[
\frac{3x^2 + 18x + 27}{x(x + 3)^2} - \frac{6x^2 + 18x}{x(x + 3)^2} - \frac{x^2}{x(x + 3)^2}
\]

### Step 4: Combine the numerators
Now, combine the numerators:

\[
(3x^2 + 18x + 27) - (6x^2 + 18x) - x^2
\]

Simplify:

\[
3x^2 + 18x + 27 - 6x^2 - 18x - x^2 = -4x^2 + 27
\]

### Step 5: Final expression
The final simplified expression is:

\[
\frac{-4x^2 + 27}{x(x + 3)^2}
\]

Would you like any further details on this process or have any questions?

### 5 Related Questions:
1. How do you find the least common denominator for more complex fractions?
2. What is the difference between simplifying a fraction and factoring it?
3. How do you expand and simplify expressions involving binomials like \( (x+3)^2 \)?
4. Can you explain how to handle negative exponents in similar expressions?
5. How can you check if a rational expression can be simplified further?

### Tip:
When simplifying rational expressions, always look for common factors in the numerator and denominator to potentially reduce the expression further.

Simplify the expression: \frac{3}{x} - \frac{6}{x + 3} - \frac{x}{(x + 3)^2}.

Learn how to simplify the rational expression \frac{3}{x} - \frac{6}{x + 3} - \frac{x}{(x + 3)^2} by finding the least common denominator and combining terms. This problem involves algebraic concepts such as rational expressions and factoring, suitable for Grades 9-12.

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