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.