[(A+b)^2+a^2-b^2-(a-b)^2]/2a hint 2a=a+b+a-b,a^2+b^2=(a+b)(a-b),try to use x^3-y^3, idenity

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!

[(A+b)^2 + a^2 - b^2 - (a-b)^2]/2a hint 2a=a+b+a-b,a^2+b^2=(a+b)(a-b),try to use x^3-y^3 identity

Explore a detailed step-by-step solution to simplify the expression [(A + b)^2 + a^2 - b^2 - (a - b)^2]/2a. This problem uses key algebraic identities such as difference of squares and cubic factorization. Suitable for students in Grades 10-12.

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