Write down the information in the form of algebraic expression and simplify.

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 ?

#### Solution

Lenght of the rectangular farm = (2a^{2} + 3b^{2}) m

Breadth of the rectangular farm = (a^{2} + b^{2}) m

Side of the square plot = (a^{2} − b^{2}) m

∴ Area of the remaining part of the farm

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

= Lenght of the rectangular farm × Breadth of the rectangular farm − (Side of the square plot)^{2}

= (2a^{2} + 3b^{2}) × (a^{2} + b^{2}) − (a^{2} − b^{2})^{2}

= 2a^{2}(a^{2} + b^{2}) + 3b^{2}(a^{2} + b^{2}) − (a^{4} + b^{4 }− 2a^{2}b^{2})

= 2a^{4 }+ 2a^{2}b^{2 }+^{ }3a^{2}b^{2} + 3b^{4} − a^{4} − b^{4 }+ 2a^{2}b^{2}

= 2a^{4} − a^{4}^{ }+ 2a^{2}b^{2 }+^{ }3a^{2}b^{2 }+ 2a^{2}b^{2 }+ 3b^{4 }− b^{4}

= (a^{4} + 7a^{2}b^{2 }+ 2b^{4}) m^{2}

Thus, the area of the remaining part of the farm is (a^{4} + 7a^{2}b^{2 }+ 2b^{4}) m^{2}.