Question

There is a rectangular farm with length `(2a^2 + 3b^2)` metre and breadth `(a^2 + b^2)` metre. The farmer used a square shaped plot of the farm to build a house. The side of the plot was `(a^2 -  b^2)` metre.
What is the area of the remaining part of the farm?

Answer 1

Lenght of the rectangular farm = (2a² + 3b²) m

Breadth of the rectangular farm = (a² + b²) m

The total area of the farm = Length of the rectangular farm × Breadth of the rectangular farm

= (2a² + 3b²) × (a² + b²)

= 2a²(a² + b²) + 3b²(a² + b²)

= 2a⁴ + 2a²b² +3a²b² + 3b⁴ 

= (2a⁴ + 5a²b² + 3b²) sq.meter

Side of the square plot = (a² − b²) meter.

Area of the square plot = (side)²

=  (a² − b²)²

= a − 2a²b² + b⁴

 Area of the remaining part of the farm = Total area of the farm − Area of the square plot

= (2a⁴ + 5a²b² + 3b⁴) - (a⁴ − 2a²b² + b⁴)

= 2a⁴ + 5a²b² + 3b⁴ − a⁴ − 2a²b² + b⁴ 

= 2a⁴ − a⁴ + 5a²b² + 2a²b² + 3b⁴ − b⁴ 

= a⁴ + 7a²b² + 2b⁴

Thus, the area of the remaining part of the farm is (a⁴ + 7a²b² + 2b⁴) sq. meter.