4. The area of a rectangle is 180 units $^{2}$ and the perimeter is 54 units. If the length of each side of the rectangle is increased by six units, what is the area of the resulting rectangle?
Solution 1: Let $x$ and $y$ be the sides of the rectangle. Then $x y=180$ and $2 x+2 y=54$ or $x+y=27$. By inspection we can see that $x=12, y=15$ and $x=15, y=12$ are the solutions. When each side is increased by 6 , the area of the new rectangle is $(12+6)(15+6)=18 * 21=$ 378 .
Solution 2: Let $x$ and $y$ be the sides of the rectangle. From the question we know that $x y=180$ and $2 x+2 y=54$. We want to determine
$$
\begin{aligned}
(x+6)(y+6) & =x y+6 x+6 y+36 \\
& =x y+3(2 x+2 y)+36 \\
& =180+162+36 \\
& =378
\end{aligned}
$$