Simplify (x² − xy + y) − 3(x² + xy − 2y).

Step-by-step solution

1. 1

   Distribute the −3 over every term inside.

   $-3\left(x^{2}\right)=-3x^{2},\qquad -3(xy)=-3xy,\qquad -3(-2y)=+6y$

   The last product is the one most often mishandled: two negatives make the $y$ term positive.

2. 2

   Drop the first bracket unchanged and write everything out. The leading bracket has an implicit $+1$ in front, so its signs are untouched:

   $x^{2}-xy+y-3x^{2}-3xy+6y$

3. 3

   Group the three families of like terms. $x^{2}$, $xy$ and $y$ are three different types and cannot be merged with each other:

   $\left(x^{2}-3x^{2}\right)+(-xy-3xy)+(y+6y)$

4. 4

   Combine each group.

   $-2x^{2}-4xy+7y$

   $\boxed{-2x^{2}-4xy+7y}$

5. 5

   Check at two points, and note a widely printed wrong answer. At $x=1$, $y=1$: the original is $(1-1+1)-3(1+1-2)=1-0=1$, and the answer gives $-2-4+7=1$ ✓. At $x=2$, $y=1$: original $=(4-2+1)-3(4+2-2)=3-12=-9$; answer $=-8-8+7=-9$ ✓. The answer $-x^{2}-2xy-5y$ appears in some worked solutions; at $x=1,y=1$ it gives $-8$, so the substitution check rejects it at once.