1. The problem is to expand and simplify the expressions $(x+y)^2$ and $x^2-xy-2y^2$. 2. To expand $(x+y)^2$, use the formula for the square of a binomial: $$(a+b)^2 = a^2 + 2ab + b^2.$$ Apply this to $x$ and $y$: $$ (x+y)^2 = x^2 + 2xy + y^2. $$ 3. Now consider the expression $x^2 - xy - 2y^2$. We check if it can be factored. 4. Look for factors of the form $(x + ay)(x + by)$ such that: $ab = -2$ and $a + b = -1$ (the coefficient of the middle term $-xy$). 5. The pair of numbers $a=1$ and $b=-2$ works because: $$1 imes (-2) = -2$$ $$1 + (-2) = -1$$ 6. Factorization: $$ x^2 - xy - 2y^2 = (x + y)(x - 2y). $$ Final answers: $$(x+y)^2 = x^2 + 2xy + y^2,$$ $$(x^2 - xy - 2y^2) = (x + y)(x - 2y).$$