What is the area of the region bounded by |x| ≤ 2k and |y| ≤ k ,...

The given inequalities are $|x| \leq 2k$ and $|y| \leq k$.

This can be written as $-2k \leq x \leq 2k$ and $-k \leq y \leq k$.

These inequalities represent a rectangular region in the $xy$-plane.

The length of the rectangle along the $x$-axis is $2k - (-2k) = 4k$.

The width of the rectangle along the $y$-axis is $k - (-k) = 2k$.

The area of the rectangular region is given by length \times width.

Area $= 4k \times 2k = 8k^2$.

Answer: $8k^2$