Step 1: Expand the Expressions
To simplify (x+y)(2x+y)+(x+2y)(x-y), we first need to expand both products.
- For (x+y)(2x+y):
- Distribute each term in the first bracket by each term in the second bracket.
- This results in:
- x * 2x = 2x²
- x * y = xy
- y * 2x = 2xy
- y * y = y²
- Combine these results:
- 2x² + xy + 2xy + y² = 2x² + 3xy + y²
- For (x+2y)(x-y):
- Again, distribute each term in the first bracket by each term in the second bracket.
- This results in:
- x * x = x²
- x * (-y) = -xy
- 2y * x = 2xy
- 2y * (-y) = -2y²
- Combine these results:
- x² - xy + 2xy - 2y² = x² + xy - 2y²
Step 2: Combine the Results
Now, we need to combine the two expanded expressions:
- From (x+y)(2x+y): 2x² + 3xy + y²
- From (x+2y)(x-y): x² + xy - 2y²
Adding these together gives:
- 2x² + x² + 3xy + xy + y² - 2y²
Step 3: Simplify the Combined Expression
Now, combine like terms:
- For x² terms: 2x² + x² = 3x²
- For xy terms: 3xy + xy = 4xy
- For y² terms: y² - 2y² = -y²
Final Result
The simplified expression is:
3x² + 4xy - y²