The problem asks for the area of a rectangle with given length and breadth.

• Length $L = 2x^2 + 5xy + 5y^2$
• Breadth $B = x^2 - xy + 6y^2$

The area $A$ of a rectangle is calculated as:

$A = \text{Length} \times \text{Breadth}$

Now, we need to multiply these two expressions:

$A = (2x^2 + 5xy + 5y^2) \times (x^2 - xy + 6y^2)$

I'll expand this product step-by-step.

Step 1: Distribute the terms from the first expression over the second

$A = 2x^2(x^2 - xy + 6y^2) + 5xy(x^2 - xy + 6y^2) + 5y^2(x^2 - xy + 6y^2)$

Step 2: Multiply each term

1. $2x^2(x^2 - xy + 6y^2) = 2x^4 - 2x^3y + 12x^2y^2$
2. $5xy(x^2 - xy + 6y^2) = 5x^3y - 5x^2y^2 + 30xy^3$
3. $5y^2(x^2 - xy + 6y^2) = 5x^2y^2 - 5xy^3 + 30y^4$

Step 3: Combine all terms

$A = 2x^4 - 2x^3y + 12x^2y^2 + 5x^3y - 5x^2y^2 + 30xy^3 + 5x^2y^2 - 5xy^3 + 30y^4$

Step 4: Simplify by combining like terms

$A = 2x^4 + (5x^3y - 2x^3y) + (12x^2y^2 - 5x^2y^2 + 5x^2y^2) + (30xy^3 - 5xy^3) + 30y^4$

$A = 2x^4 + 3x^3y + 12x^2y^2 + 25xy^3 + 30y^4$

Thus, the area of the rectangle is:

$A = 2x^4 + 3x^3y + 12x^2y^2 + 25xy^3 + 30y^4$

Would you like more details on any part of this solution? Here are some related questions for further exploration:

1. How do you expand and simplify polynomial expressions?
2. What are some other methods for multiplying polynomials?
3. What is the geometric interpretation of multiplying these polynomials?
4. Can this expression be factored further?
5. How does this concept apply to other shapes, like triangles or circles?

Tip: Always group like terms when expanding polynomial expressions to simplify them efficiently.