The length of a rectangle is increased by 3 m and its breadth is decreased by 2 m…

Step 1. First fact. $(x + 3)(y - 2) = xy$ gives $xy - 2x + 3y - 6 = xy$, so $-2x + 3y = 6$.

Step 2. Second fact. $(x - 2)(y + 3) = xy + 10$ gives $xy + 3x - 2y - 6 = xy + 10$, so $3x - 2y = 16$.

Step 3. Solve. Multiply the first equation by $3$ and the second by $2$: $-6x + 9y = 18$ and $6x - 4y = 32$. Add: $5y = 50$, so $y = 10$. Then $3x = 16 + 20 = 36$, so $x = 12$.

Step 4. The perimeter. $2(12 + 10) = 44\ \text{m}$.