Simplify 2(x²y + xy) − 3(x²y − xy) + 2xy.

Step-by-step solution

1. 1

   Remove the brackets one at a time, tracking the sign of each multiplier. The first bracket is multiplied by $+2$ and the second by $-3$. It is the $-3$ that causes trouble: it multiplies both terms inside, and $-3\times(-xy)=+3xy$, not $-3xy$.

2. 2

   Expand the first bracket.

   $2\left(x^{2}y+xy\right)=2x^{2}y+2xy$

3. 3

   Expand the second bracket.

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

4. 4

   Write out all the terms.

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

5. 5

   Collect the two families of like terms. $x^{2}y$ and $xy$ are not like terms (different powers of $x$), so they stay separate:

   $(2-3)x^{2}y+(2+3+2)xy=-x^{2}y+7xy$

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

6. 6

   Check numerically, and note the common wrong answer. At $x=2$, $y=1$: the original is $2(4+2)-3(4-2)+4=12-6+4=10$, and the answer gives $-4+14=10$ ✓. A frequently seen incorrect result is $-x^{2}y-xy+2xy=-x^{2}y+xy$, which comes from writing $-3xy$ instead of $+3xy$ when distributing; it would give $-4+2=-2$ at the same test point, so the substitution check catches it immediately.