The expression you provided is:

\[
(a+b)^2 (a-b)^2 + 1
\]

We can simplify this step by step.

### Step 1: Expand \((a+b)^2\) and \((a-b)^2\).

- \((a+b)^2 = a^2 + 2ab + b^2\)
- \((a-b)^2 = a^2 - 2ab + b^2\)

### Step 2: Multiply the expanded forms.

Now, multiply the two expanded binomials:

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

This is a product of sums and differences, which can be simplified using the identity \((x + y)(x - y) = x^2 - y^2\).

Here, \(x = a^2 + b^2\) and \(y = 2ab\). Applying the identity:

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

### Step 3: Simplify the expression.

- \((a^2 + b^2)^2 = a^4 + 2a^2b^2 + b^4\)
- \((2ab)^2 = 4a^2b^2\)

Now, subtract the two:

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

### Step 4: Add 1 to the result.

The final expression is:

\[
a^4 - 2a^2b^2 + b^4 + 1
\]

Thus, the simplified form of \((a+b)^2 (a-b)^2 + 1\) is:

\[
a^4 - 2a^2b^2 + b^4 + 1
\]

Would you like a detailed explanation of any step?

Here are 5 related questions to further your understanding:
1. What is the significance of the identity \((x + y)(x - y) = x^2 - y^2\)?
2. How can we factorize the expression \(a^4 - 2a^2b^2 + b^4\)?
3. What are the applications of binomial expansions in higher mathematics?
4. How does the difference of squares formula help in simplifications?
5. How can we apply this method to simplify higher degree polynomials?

**Tip:** When dealing with products of binomials, always check if any algebraic identity can simplify the expression before expanding everything.

This page explains the simplification of the expression (a+b)² (a-b)² + 1 step-by-step, covering the binomial expansion and the difference of squares identity. Suitable for students in Grades 9-12.

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