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The given expression is:

\[
(x + 4y)^2 - (x + y)(x - y)
\]

Now, let's simplify it step by step.

### Step 1: Expand \((x + 4y)^2\)
\[
(x + 4y)^2 = x^2 + 8xy + 16y^2
\]

### Step 2: Expand \((x + y)(x - y)\) (difference of squares)
\[
(x + y)(x - y) = x^2 - y^2
\]

### Step 3: Substitute both expanded forms into the original expression
\[
x^2 + 8xy + 16y^2 - (x^2 - y^2)
\]

### Step 4: Simplify the expression by combining like terms
\[
x^2 + 8xy + 16y^2 - x^2 + y^2 = 8xy + 17y^2
\]

Thus, the simplified expression is:
\[
8xy + 17y^2
\]

Would you like any further details or explanations?

Here are some related questions to explore:
1. How can we generalize binomial expansion for \((a + b)^n\)?
2. What are the steps to simplify algebraic expressions involving multiple variables?
3. How does the difference of squares help simplify expressions?
4. Can you verify the simplification using factoring techniques?
5. How would this expression behave if \(y = 0\)?

**Tip:** Always check if common factors can be factored out to further simplify expressions.

Learn how to simplify the algebraic expression (x + 4y)^2 - (x + y)(x - y) using polynomial expansion and the difference of squares. This problem explores key concepts like algebra and polynomial simplification, suitable for Grades 9-11.

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