**Solution:**

To find the number of scalene triangles with integral sides that can be formed with a perimeter of 45 cm, we need to consider the conditions for a triangle to be scalene and have integral sides.

**Conditions for a scalene triangle:**
1. All three sides must have different lengths.
2. The sum of the lengths of any two sides must be greater than the length of the third side.

**Approach:**
We will consider the possible values for the lengths of the sides of the triangle and check if they satisfy the conditions for a scalene triangle.

**Step 1:**
Let's start by assuming the three sides of the triangle as a, b, and c.

**Step 2:**
As the perimeter is given as 45 cm, we have the equation: a + b + c = 45.

**Step 3:**
To satisfy the condition of a scalene triangle, the sides a, b, and c must have different lengths. Therefore, we can assume the following conditions:

- a < b="" />< />

**Step 4:**
We will consider the possible values for 'a' and check the range of values for 'b' and 'c' that satisfy the conditions.

**Step 5:**
The maximum possible value for 'a' can be (45 - 1 - 1) = 43, as 'b' and 'c' should have lengths greater than 'a'.

**Step 6:**
For each value of 'a' from 1 to 43, we will check the range of values for 'b' and 'c' that satisfy the conditions.

**Step 7:**
For a given value of 'a', the minimum possible value for 'b' can be (a + 1).

**Step 8:**
To satisfy the condition a + b + c = 45, the maximum possible value for 'b' can be (45 - a - 1).

**Step 9:**
We will check the range of values for 'b' (from minimum to maximum) for each value of 'a' and count the number of combinations that satisfy the conditions.

**Step 10:**
By iterating through the possible values of 'a' and counting the number of combinations, we find that there are a total of 37 scalene triangles with integral sides that can be formed with a perimeter of 45 cm.