Explanation:
When the area of a rectangle increases, it means that either the length or the width or both have increased. Let's assume that the length and width of the rectangle are l and w respectively.

- Area of rectangle = length x width (A = lw)
- Perimeter of rectangle = 2(length + width) (P = 2(l + w))

Now, given that the area increases from 2 cm2 to 4 cm2.

- Initially, A1 = lw = 2
- After the increase in area, A2 = lw = 4

We need to find out what happens to the perimeter.

- Initially, P1 = 2(l + w)
- After the increase in area, we don't know the new values of l and w, so let's assume that they are l' and w' respectively.
- P2 = 2(l' + w')

To solve this problem, we need to use the concept of factorization. We know that:

- A1 = lw = 2 = 1 x 2
- A2 = lw = 4 = 2 x 2

Therefore, the factors of A1 are 1 and 2, and the factors of A2 are 2 and 2.

Now, we need to find out the possible values of l and w for each case.

- Case 1: A1 = lw = 2

The factors of 2 are 1 and 2. So, the possible values of l and w are:

- l = 1, w = 2
- l = 2, w = 1

In both cases, the perimeter is:

- P1 = 2(l + w) = 2(1 + 2) = 6

- Case 2: A2 = lw = 4

The factors of 4 are 1 and 4, and 2 and 2. So, the possible values of l and w are:

- l = 1, w = 4
- l = 4, w = 1
- l = 2, w = 2

In the first two cases, the perimeter is:

- P2 = 2(l + w) = 2(1 + 4) = 10

In the third case, the perimeter is:

- P2 = 2(l + w) = 2(2 + 2) = 8

Therefore, we can see that when the area of a rectangle increases from 2 cm2 to 4 cm2:

- The perimeter can increase from 6 cm to 8 cm or 10 cm, depending on the values of length and width.
- In general, the perimeter will increase if the length and width both increase, or if one of them increases significantly while the other remains almost the same.
- So, the correct answer is option A: the perimeter will increase.

If the area will increase than its sides will increase which results in the increase in perimeter