https://file.aiquickdraw.com/mathbot/file/1727216516321loxkapoi.jpg

The expression in the image is a mathematical fraction with two different parts in the numerator and the denominator.

The numerator is:  
\[
(a + b)^2 - c^2
\]
The denominator is:
\[
(c - a)^2 - b^2
\]

### Step-by-Step Solution:

#### 1. Simplify the Numerator
The numerator is \((a + b)^2 - c^2\), which is a **difference of squares**. We can factor this as:
\[
(a + b)^2 - c^2 = [(a + b) - c][(a + b) + c]
\]
So the numerator becomes:
\[
(a + b - c)(a + b + c)
\]

#### 2. Simplify the Denominator
The denominator is \((c - a)^2 - b^2\), which is again a **difference of squares**. We can factor this as:
\[
(c - a)^2 - b^2 = [(c - a) - b][(c - a) + b]
\]
So the denominator becomes:
\[
(c - a - b)(c - a + b)
\]

### The Final Simplified Expression:
Thus, the given expression simplifies to:
\[
\frac{(a + b - c)(a + b + c)}{(c - a - b)(c - a + b)}
\]

Would you like further details or have any questions?

Here are 5 related questions that expand on this topic:
1. How do we handle the difference of squares when factoring more complex polynomials?
2. Can the expression be further simplified under specific values of \(a\), \(b\), or \(c\)?
3. What are the common factoring techniques for quadratic expressions?
4. How does recognizing the structure of an equation help in algebraic manipulations?
5. How is the difference of squares used in real-world applications of algebra?

**Tip:** Always look for patterns like the difference of squares, as it helps in factoring and simplifying expressions quickly!

Simplify algebraic expressions like (a + b)^2 - c^2 and (c - a)^2 - b^2 using the difference of squares. This step-by-step solution teaches key algebraic techniques and is ideal for students in Grades 9-12.

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