QUESTION

# Give possible expressions for the length and breadth of the rectangle whose area is: $4{{a}^{2}}+\text{ }4a - 3$ .A) (2a – 3) (2a + 2)B) (2a – 1) (2a – 3)C) (2a + 3) (2a – 1)D) None of the above

Hint: Let us first learn about Rectangle and its Area.
RECTANGLE: A parallelogram all of whose angles are right angles, especially the one with adjacent sides of unequal length is called a rectangle.
AREA OF RECTANGLE: The formula that is used to calculate the area of a rectangle is $L\times B$, where ‘L’ stands for the Length of the Rectangle and ‘B’ stands for Breadth of the Rectangle.

Let us now solve this question.
A) (2a– 3) (2a + 2). As we know that this is the length and breadth of the rectangle, and the formula to calculate the area of the rectangle, therefore, we will multiply (2a– 3) and (2a + 2) and check whether their product is equal to $4{{a}^{2}}+\text{ }4a - 3$ or not.
(2a– 3) (2a + 2) = 2a (2a + 2) – 3 (2a + 2) = $4{{a}^{2}}+\text{ }4a\text{ }\text{ }-6a\text{ }\text{ }-6\text{ }=\text{ }4{{a}^{2}}\text{ }-2a\text{ }\text{ }-6$
We can observe that $4{{a}^{2}}\text{ }-2a\text{ }\text{ }-6\ne 4{{a}^{2}}+\text{ }4a\text{ }\text{ }-3$ .
Therefore, this is not a correct option.
B) (2a – 1) (2a – 3). We will multiply (2a– 1) and (2a – 3) and check whether their product is equal to $4{{a}^{2}}+\text{ }4a - 3$ or not.
(2a – 1) (2a – 3) = 2a (2a – 3) – 1 (2a – 3) = $4{{a}^{2}}\text{ }-6a\text{ }\text{ }-2a\text{ }+\text{ }3\text{ }=\text{ }4{{a}^{2}}\text{ }-8a\text{ }+\text{ }3$ .
We can observe that $4{{a}^{2}}\text{ }-8a\text{ }+\text{ }3\ne 4{{a}^{2}}+\text{ }4a\text{ }\text{ }-3$ .
Therefore, this is not a correct option.
C) (2a + 3) (2a – 1). We will multiply (2a + 3) and (2a – 1) and check whether their product is equal to $4{{a}^{2}}+\text{ }4a - 3$ or not.
(2a + 3) (2a – 1) = 2a (2a – 1) + 3 (2a – 1) = $4{{a}^{2}}\text{ }-2a\text{ }+\text{ }6a\text{ }\text{ }-3\text{ }=\text{ }4{{a}^{2}}+\text{ }4a\text{ }\text{ }-3$ .
We can observe that $4{{a}^{2}}+\text{ }4a\text{ }\text{ }-3=4{{a}^{2}}+\text{ }4a\text{ }\text{ }-3$. Therefore, this is the correct option.
D) None of the above. This cannot be the correct option as the correct option is (C).
After solving this question, the answer of this question is (C) (2a + 3) (2a – 1).

Note: One must do all the calculation very carefully in this question. Any mistake in the calculations can make the answer wrong.