\left(a^2+b^2\right)\left(a+b\right)\left(a-b\right)

We are given the expression:

\[
\left(a^2 + b^2\right)\left(a + b\right)\left(a - b\right)
\]

To simplify this, we can use known algebraic identities.

First, recognize that the product \((a + b)(a - b)\) is the difference of squares identity:

\[
(a + b)(a - b) = a^2 - b^2
\]

Substituting this into the expression, we get:

\[
\left(a^2 + b^2\right)\left(a^2 - b^2\right)
\]

Now, expand this product using the distributive property:

\[
(a^2 + b^2)(a^2 - b^2) = a^2(a^2 - b^2) + b^2(a^2 - b^2)
\]

Distribute the terms:

\[
= a^4 - a^2b^2 + b^2a^2 - b^4
\]

Notice that \(-a^2b^2\) and \(b^2a^2\) cancel out:

\[
a^4 - b^4
\]

Thus, the simplified expression is:

\[
a^4 - b^4
\]

### Final answer:
\[
a^4 - b^4
\]

Would you like more details on the steps involved or have any questions?

Here are five related questions:
1. How does the difference of squares identity work?
2. What happens if \(a = b\) in this expression?
3. Can this expression be factored further if \(a^4 - b^4\) was treated as a difference of squares?
4. How does this process change if there were higher powers of \(a\) and \(b\)?
5. How would the result differ if the original expression had \(a^2 - b^2\) instead of \(a^2 + b^2\)?

Tip: Always look for common algebraic identities like the difference of squares to simplify complex expressions.

Learn how to simplify the algebraic expression (a^2 + b^2)(a + b)(a - b) using the difference of squares identity and polynomial expansion. This step-by-step solution covers essential algebra concepts, including the difference of squares and distributive property, and is suitable for students in Grades 9-11.

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