Question: User: top-left: $6x + 3 + x + 6$
top-left: $7x + 9$
f) $2x - x(y + x) + (x + y)(2x - 2y) =$
$2x - xy - x^2 + 2x^2 + 2xy + 2xy - 2y^2$
$2x - 5xy + 3x^2 - 2y^2$
1. **State the problem:** Simplify the expression $$2x - x(y + x) + (x + y)(2x - 2y)$$.
2. **Apply distributive property:**
$$2x - x(y + x) + (x + y)(2x - 2y) = 2x - (xy + x^2) + (x + y)(2x - 2y)$$
3. **Distribute inside the second parentheses:**
$$(x + y)(2x - 2y) = x(2x - 2y) + y(2x - 2y) = 2x^2 - 2xy + 2xy - 2y^2$$
4. **Simplify the terms inside the parentheses:**
$$2x^2 - 2xy + 2xy - 2y^2 = 2x^2 + \cancel{- 2xy + 2xy} - 2y^2 = 2x^2 - 2y^2$$
5. **Substitute back:**
$$2x - xy - x^2 + 2x^2 - 2y^2$$
6. **Combine like terms:**
$$- x^2 + 2x^2 = x^2$$
7. **Final simplified expression:**
$$2x - xy + x^2 - 2y^2$$
**Note:** The user's intermediate step $2x - 5xy + 3x^2 - 2y^2$ is incorrect because the $xy$ terms were not combined properly.
**Answer:** $$2x - xy + x^2 - 2y^2$$