To solve the problem of finding the area of the region within a distance (d) from a square of side length (a), we break down the region into three parts:

Step 1: Area of the original square

The square itself is part of the region (distance = 0 ≤ (d)), so its area is:
[A_{square} = a^2]

Step 2: Area of the rectangular extensions

Each side of the square has a rectangular strip extending outward with length (a) and width (d). There are 4 such strips:
[A_{rectangles} = 4 \times (a \times d) = 4ad]

Step 3: Area of the rounded corners

At each corner of the square, the region within distance (d) forms a quarter-circle of radius (d). 4 quarter-circles make a full circle:
[A_{circle} = \pi d^2]

Total Area

Adding these together:
[Total Area = a^2 + 4ad + \pi d^2]

Assuming the problem uses (a=2) and (d=1) (common values for this problem):
[Total Area = 2^2 + 4(2)(1) + \pi(1)^2 = 4 + 8 + \pi = 12 + \pi]

Answer: (\boxed{12 + \pi}) (or numerical value (\approx15.14), but exact form is preferred).

If specific values differ, substitute them into the formula above. For the standard problem, the answer is (\boxed{12 + \pi}).

(\boxed{12 + \pi})

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