25a^2 - 35a + 12
Split the middle term..
=25a^2 - (15+20)a + 12
=25a^2 - 15a-20a + 12
=5a(5a-3) -4(5a-3)
=(5a-3) (5a-4)...

Expression for the Length and Breadth of a Rectangle

In order to find the possible expressions for the length and breadth of a rectangle given its area, we need to factorize the given area expression and identify the factors that can represent the length and breadth.

Factorizing the Area Expression
The given area expression is 25a² - 35a + 12. To factorize this expression, we need to find two binomials that multiply together to give the original expression. We can use various methods like trial and error, grouping, or quadratic formula to factorize the expression. In this case, we can use trial and error to find the factors.

Step 1: Multiply the coefficients of the square term and the constant term: 25 * 12 = 300.

Step 2: Find two numbers that multiply to give 300 and add up to the coefficient of the linear term (-35). In this case, the numbers are -20 and -15 since -20 * -15 = 300 and -20 + (-15) = -35.

Step 3: Rewrite the expression using the numbers found in the previous step: 25a² - 20a - 15a + 12.

Step 4: Group the terms: (25a² - 20a) + (-15a + 12).

Step 5: Factor out the common factors from each group: 5a(5a - 4) - 3(5a - 4).

Step 6: Notice that we have a common binomial factor (5a - 4). We can factor it out: (5a - 4)(5a - 3).

Expression for Length and Breadth
From the factorization, we can determine the possible expressions for the length and breadth of the rectangle.

The length (L) can be represented by the factor (5a - 4), and the breadth (B) can be represented by the factor (5a - 3).

Therefore, the possible expressions for the length and breadth of the rectangle are:
- Length (L) = 5a - 4
- Breadth (B) = 5a - 3

Explanation:
The factorization process helps us identify the expression for the length and breadth by breaking down the area expression into its factors. By factoring out the common binomial factor, we can determine the expressions that represent the length and breadth of the rectangle. These expressions allow us to calculate the length and breadth for any given value of 'a' and help us understand the relationship between the area and the dimensions of the rectangle.