A rectangular field has an area `(14x^2 - 11x - 15) m^2`. What could be the possible expression for length and breadth of the field?

Options

A. (3x - 2) m and (5x + 8) m
B. (7x + 5) m and (2x - 3) m
C. (4x + 2) m and (3x - 7) m
D. (7x - 5) m and (2x - 3) m

Answer

To find the possible expressions for the length and breadth of a rectangular field given its area, we need to factor the quadratic expression for the area. The area is given as:

\[ A = 14x^2 - 11x - 15 \, m^2 \]

### Step 1: Factor the quadratic expression
We will factor the quadratic expression \( 14x^2 - 11x - 15 \).

To factor this, we look for two numbers that multiply to \( 14 \times (-15) = -210 \) and add up to \( -11 \).

The numbers that satisfy this condition are \( 21 \) and \( -10 \) because:

\[ 21 \times (-10) = -210 \]
\[ 21 + (-10) = 11 \]

### Step 2: Rewrite the middle term
We can rewrite the expression by splitting the middle term using \( 21 \) and \( -10 \):

\[ 14x^2 + 21x - 10x - 15 \]

### Step 3: Group the terms
Now, we group the terms:

\[ (14x^2 + 21x) + (-10x - 15) \]

### Step 4: Factor by grouping
Next, we factor out the common factors from each group:

1. From the first group \( 14x^2 + 21x \), we can factor out \( 7x \):
\[ 7x(2x + 3) \]

2. From the second group \( -10x - 15 \), we can factor out \( -5 \):
\[ -5(2x + 3) \]

Now we have:

\[ 7x(2x + 3) - 5(2x + 3) \]

### Step 5: Factor out the common binomial
Now we can factor out the common binomial \( (2x + 3) \):

\[ (2x + 3)(7x - 5) \]

### Conclusion
Thus, the area \( 14x^2 - 11x - 15 \) can be expressed as the product of two factors:

- Length: \( (7x - 5) \)
- Breadth: \( (2x + 3) \)

### Final Answer
The possible expressions for the length and breadth of the field are:

- Length: \( 7x - 5 \)
- Breadth: \( 2x + 3 \)