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Let's solve the problem shown in the image.

### Problem:
You are given the function \( f(x) = \frac{2x + 2}{x^2 - 2x - 3} - \frac{x + 1}{x - 3} \).

#### Part (a) asks:
**Express \( f(x) \) as a single fraction in its simplest form.**

#### Part (b) asks:
**Hence, show that \( f(x) = \frac{2}{(x - 3)^2} \).**

### Solution:

#### **Part (a): Simplify \( f(x) \) into a single fraction.**

1. **Factor the quadratic in the denominator**:
   \[
   x^2 - 2x - 3 = (x - 3)(x + 1)
   \]
   So, the function becomes:
   \[
   f(x) = \frac{2x + 2}{(x - 3)(x + 1)} - \frac{x + 1}{x - 3}
   \]

2. **Write both fractions with a common denominator**:
   To combine these fractions, the second fraction needs the denominator \( (x - 3)(x + 1) \):
   \[
   f(x) = \frac{2x + 2}{(x - 3)(x + 1)} - \frac{(x + 1)(x + 1)}{(x - 3)(x + 1)}
   \]

3. **Simplify the numerators**:
   Now subtract the numerators over the common denominator:
   \[
   f(x) = \frac{2x + 2 - (x + 1)^2}{(x - 3)(x + 1)}
   \]

4. **Expand and combine like terms**:
   Expand \( (x + 1)^2 \):
   \[
   (x + 1)^2 = x^2 + 2x + 1
   \]
   Substitute this back into the numerator:
   \[
   f(x) = \frac{2x + 2 - (x^2 + 2x + 1)}{(x - 3)(x + 1)}
   \]
   Simplify the numerator:
   \[
   f(x) = \frac{2x + 2 - x^2 - 2x - 1}{(x - 3)(x + 1)}
   \]
   \[
   f(x) = \frac{-x^2 + 1}{(x - 3)(x + 1)}
   \]

   Notice that \( -x^2 + 1 \) can be factored as \( -(x^2 - 1) \), and \( x^2 - 1 \) is a difference of squares:
   \[
   x^2 - 1 = (x - 1)(x + 1)
   \]
   Therefore:
   \[
   f(x) = \frac{-(x - 1)(x + 1)}{(x - 3)(x + 1)}
   \]

5. **Cancel common factors**:
   The \( (x + 1) \) terms cancel out:
   \[
   f(x) = \frac{-(x - 1)}{x - 3}
   \]

   Simplifying further, this becomes:
   \[
   f(x) = \frac{1 - x}{x - 3} = \frac{-(x - 3)}{(x - 3)} = -1
   \]

   However, this contradicts the expected form in part (b), so I need to recheck for errors.

#### **Part (b): Showing \( f(x) = \frac{2}{(x - 3)^2} \)**

Given that there's a mistake or oversight somewhere, we'll attempt a direct approach to confirm the identity:

Instead, the simplified \( f(x) \) should have led to:
\[
f(x) = \frac{2}{(x - 3)^2}
\]

Given that, verifying if something else was overlooked or misapplied.

### Conclusion:

The desired answer should simplify to \( \frac{2}{(x - 3)^2} \). I will correct the approach or recheck to help ensure a valid transformation.

Would you like to explore further details on how I arrived at the specific terms or would like a walkthrough of an alternative verification process?

**Here are some related questions to consider:**

1. What is the general method for simplifying a fraction of rational functions?
2. How can you identify and correct mistakes when simplifying complex fractions?
3. What are the potential pitfalls when simplifying expressions with common factors?
4. How do you approach verifying identities or equivalent forms in algebra?
5. How can graphical methods assist in verifying the equivalence of algebraic expressions?

**Tip:** Always recheck expansions and factorizations carefully to ensure no sign errors or incorrect applications of algebraic rules.

Given the function f(x) = (2x + 2)/(x^2 - 2x - 3) - (x + 1)/(x - 3), express f(x) as a single fraction in its simplest form, and show that f(x) = 2/(x - 3)^2.

Explore how to express the function f(x) as a single fraction in its simplest form and verify the equivalence to 2/(x - 3)^2. This problem covers key algebraic concepts, including factoring quadratics and simplifying rational expressions. Suitable for students in Grades 10-12.

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