Let's solve the expression step by step, using the provided hints:

The expression is:

$\frac{(A + b)^2 + a^2 - b^2 - (a - b)^2}{2a}$

Step 1: Apply the identity for $2a = (a+b) + (a-b)$

This identity will help simplify the denominator: $2a = (a+b) + (a-b)$ This means the denominator is $(a+b) + (a-b)$.

Step 2: Use the identity for $a^2 - b^2$

Now, apply the difference of squares identity to simplify terms like $a^2 - b^2$: $a^2 - b^2 = (a+b)(a-b)$

Step 3: Expand and simplify the numerator

The numerator is: $(A + b)^2 + a^2 - b^2 - (a - b)^2$ Let's expand each term:

• $(A + b)^2 = A^2 + 2Ab + b^2$
• $(a - b)^2 = a^2 - 2ab + b^2$

Substitute these into the numerator: $A^2 + 2Ab + b^2 + a^2 - b^2 - (a^2 - 2ab + b^2)$ Simplifying, we get: $A^2 + 2Ab + b^2 + a^2 - b^2 - a^2 + 2ab - b^2$ Now cancel terms: $A^2 + 2Ab + 2ab$ This simplifies further to: $A^2 + 2(A + a)b$

Step 4: Final expression

Now, the original expression becomes: $\frac{A^2 + 2(A + a)b}{(a+b) + (a-b)}$ Since the denominator is $2a$, the expression is: $\frac{A^2 + 2(A + a)b}{2a}$

Step 5: Consider using the identity $x^3 - y^3$

At this point, it's unclear how directly the identity $x^3 - y^3$ applies here without further transformation of the expression. However, this identity might be useful in cases where you factor more complicated terms.

---

Would you like to go into more detail on any of these steps or explore further simplifications? Here are some relative questions to explore:

1. How does the identity $x^3 - y^3$ factor into more complex expressions?
2. Could we apply any other factoring techniques to simplify this further?
3. What conditions on $A$, $a$, or $b$ might simplify the expression further?
4. How could we interpret this result geometrically or in another context?
5. Can we explore other identities that simplify quadratic expressions like this?

Tip: Always double-check identities before applying them to ensure they align with the structure of your expression!