The expression provided is:

$(x + 4y)^2 - (x - 5y)^2$

This is a difference of squares, and it can be factored using the identity:

$a^2 - b^2 = (a - b)(a + b)$

Here:

• $a = (x + 4y)$
• $b = (x - 5y)$

Step 1: Apply the formula

Using the difference of squares formula, we expand it as:

[ (x + 4y)^2 - (x - 5y)^2 = \big[(x + 4y) - (x - 5y)\big] \cdot \big[(x + 4y) + (x - 5y)\big] ]

Step 2: Simplify each term

First term: $(x + 4y) - (x - 5y)$

Distribute the negative sign inside the parentheses: $(x + 4y) - (x - 5y) = x + 4y - x + 5y = 9y$

Second term: $(x + 4y) + (x - 5y)$

Add the terms together: $(x + 4y) + (x - 5y) = x + 4y + x - 5y = 2x - y$

Step 3: Combine the results

The expression becomes: $(x + 4y)^2 - (x - 5y)^2 = (9y)(2x - y)$

Final Answer:

$(x + 4y)^2 - (x - 5y)^2 = 9y(2x - y)$

Would you like further explanation or details?

5 Related Questions:

1. How do you expand binomials using the square formula $(a + b)^2$?
2. Can you explain the concept of the difference of squares in more depth?
3. How would the expression change if the middle sign were $+$ instead of $-$?
4. How do you apply factoring techniques to higher-degree polynomials?
5. What is the geometric interpretation of the difference of squares?

Tip:

When solving algebraic expressions, always look for patterns like the difference of squares or perfect square trinomials—they often simplify the problem significantly!