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

2 min read Jun 16, 2024

Factoring and Simplifying (x+2y)^2 - 4(2x-y)^2

This expression involves squaring binomials and then subtracting the results. We can simplify this by using the following steps:

1. Expand the Squares

(x+2y)^2: This is a perfect square trinomial. We can use the formula (a+b)^2 = a^2 + 2ab + b^2.
- Applying this, we get: x^2 + 4xy + 4y^2
(2x-y)^2: Another perfect square trinomial. Applying the same formula but with subtraction, we get: 4x^2 - 4xy + y^2

2. Substitute the Expanded Forms

Now, we can substitute these expanded forms back into the original expression:

(x^2 + 4xy + 4y^2) - 4(4x^2 - 4xy + y^2)

3. Distribute the -4

Multiply the -4 by each term inside the parentheses:

x^2 + 4xy + 4y^2 - 16x^2 + 16xy - 4y^2

4. Combine Like Terms

Finally, combine all the terms with the same variables and exponents:

(-16x^2 + x^2) + (16xy + 4xy) + (4y^2 - 4y^2)

5. Simplify

This simplifies to:

-15x^2 + 20xy

Therefore, the simplified form of (x+2y)^2 - 4(2x-y)^2 is -15x^2 + 20xy.