Simplify 6xy^2 - 3xy^2 - 12xy^2.

Step-by-step solution

1. 1

   Confirm the terms are alike. Every term has the same variable part $xy^{2}$ — the same letters raised to the same powers ($x$ to the first, $y$ to the second). That is the exact condition for combining, and it means the variable part is carried along untouched while only the numbers interact.

2. 2

   Factor out the common variable part.

   $6xy^{2}-3xy^{2}-12xy^{2}=(6-3-12)\,xy^{2}$

   This is the distributive law in reverse, and it reduces the problem to a single arithmetic calculation.

3. 3

   Add the coefficients left to right. Order matters because subtraction is not associative:

   $6-3=3,\qquad3-12=-9$

   so the coefficient is $-9$. Computing $3-12$ as $12-3=9$ and reporting $+9xy^{2}$ is the usual sign slip here.

4. 4

   Write the simplified expression.

   $-9xy^{2}$

   The exponent on $y$ stays at $2$ — adding like terms never changes the powers, only the count of them.

5. 5

   Verify numerically. At $x=2$, $y=3$: the original is $6(2)(9)-3(2)(9)-12(2)(9)=108-54-216=-162$, and $-9(2)(9)=-162$ ✓. The identity holds at all $400$ integer pairs with $-10\le x,y\le9$ ✓.