Give possible expressions for the length and breadth of the rectangle whose area is given by $4 a^{2}+4 a-3$.

(A) Area of the rectangle is given by the polynomial $4 a^{2}+4 a-3$.
To find the length and breadth,we factorize the quadratic expression by splitting the middle term.
We need two numbers whose sum is $+4$ and whose product is $4 \times (-3) = -12$.
These two numbers are $+6$ and $-2$,since $6 + (-2) = 4$ and $6 \times (-2) = -12$.
Splitting the middle term $4 a$ as $6 a - 2 a$,we get:
$4 a^{2} + 6 a - 2 a - 3$
Grouping the terms:
$= 2 a(2 a + 3) - 1(2 a + 3)$
Factoring out the common binomial $(2 a + 3)$:
$= (2 a - 1)(2 a + 3)$
Since the area of a rectangle is defined as $\text{length} \times \text{breadth}$,the possible expressions for the dimensions are:
Length $= (2 a - 1)$ and Breadth $= (2 a + 3)$ or Length $= (2 a + 3)$ and Breadth $= (2 a - 1)$.