**Given Information:**
- The length of a rectangle exceeds its breadth by 9cm.
- If the length and breadth are each increased by 3cm, the area of the new rectangle will be 84cm² more than that of the given rectangle.

**Let's Solve the Problem Step-by-Step:**

**Step 1: Define the Variables**
- Let's assume the breadth of the rectangle is 'x' cm.
- Since the length of the rectangle exceeds its breadth by 9cm, we can say the length is 'x + 9' cm.

**Step 2: Calculate the Area of the Given Rectangle**
- The area of a rectangle is given by the formula: Area = Length × Breadth.
- In this case, the area of the given rectangle is: Area = (x + 9) × x = x² + 9x cm².

**Step 3: Calculate the New Length and Breadth**
- If the length and breadth are each increased by 3cm, the new length will be 'x + 9 + 3 = x + 12' cm, and the new breadth will be 'x + 3' cm.

**Step 4: Calculate the Area of the New Rectangle**
- The area of the new rectangle is: Area = (x + 12) × (x + 3) = x² + 15x + 36 cm².

**Step 5: Set up the Equation**
- The problem states that the area of the new rectangle is 84cm² more than that of the given rectangle.
- So, we can set up the equation: x² + 15x + 36 = x² + 9x + 84.

**Step 6: Solve the Equation**
- Simplifying the equation, we get: 6x - 48 = 0.
- Solving for 'x', we find: x = 8.

**Step 7: Calculate the Length and Breadth of the Given Rectangle**
- Using the value of 'x', we can find the length and breadth of the given rectangle:
- Length = x + 9 = 8 + 9 = 17 cm.
- Breadth = x = 8 cm.

**Step 8: Check the Solution**
- We can verify if the solution is correct by substituting the values back into the equation:
- Area of the given rectangle = 17 × 8 = 136 cm².
- Area of the new rectangle = (8 + 12) × (8 + 3) = 20 × 11 = 220 cm².
- The area of the new rectangle is indeed 84cm² more than that of the given rectangle.

Let the breadth of the rectangle be x cm. 

Then, the length of the rectangle is (x+9) cm.
So, area of rectangle = length x breadth =x(x+9)cm2
Now, length of new rectangle =(x+9+3) cm =(x+12) cm and
breadth of new rectangle =(x+3) cm.
So, area of new rectangle = length × breadth =(x+12)(x+3)cm2

According to the given condition, 
(x+12)(x+3)=x(x+9)+84

⇒x2+12x+3x+36=x2+9x+84

⇒15x+36=9x+84


⇒15x−9x=84−36

⇒6x=48⇒x=8

So, breadth of the rectangle is 8 cm and length 


=8+9=17 cm.