Given possible expression for the length and breadth of each of the following rectangles if.
(i) Area `=(x^(2)+5sqrt5x+30)` sq.unit.
(ii) Area `=(24 x^(2)-26x-8)` sq. unit

Answer

To solve the given problem, we need to find the possible expressions for the length and breadth of rectangles based on the provided areas. Let's tackle each part step by step.

### Part (i): Area = \(x^2 + 5\sqrt{5}x + 30\) sq. units

1. **Identify the area expression**:
The area of the rectangle is given as \(A = x^2 + 5\sqrt{5}x + 30\).

2. **Factor the quadratic expression**:
We need to factor this expression into two binomials. We will use the middle-term splitting method.

- The expression can be represented in the form \(ax^2 + bx + c\) where:
- \(a = 1\)
- \(b = 5\sqrt{5}\)
- \(c = 30\)

3. **Find two numbers that multiply to \(ac\) and add to \(b\)**:
- Here, \(ac = 1 \times 30 = 30\).
- We need two numbers that multiply to \(30\) and add to \(5\sqrt{5}\).
- The numbers \(3\sqrt{5}\) and \(2\sqrt{5}\) satisfy this condition since:
- \(3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}\)
- \(3\sqrt{5} \times 2\sqrt{5} = 6 \times 5 = 30\)

4. **Rewrite the expression**:
- Rewrite the quadratic as:
\[ x^2 + 3\sqrt{5}x + 2\sqrt{5}x + 30 \]

5. **Group the terms**:
- Group the first two and the last two terms:
\[ (x^2 + 3\sqrt{5}x) + (2\sqrt{5}x + 30) \]

6. **Factor by grouping**:
- Factor out the common terms:
\[ x(x + 3\sqrt{5}) + 2\sqrt{5}(x + 3\sqrt{5}) \]
- This gives:
\[ (x + 3\sqrt{5})(x + 2\sqrt{5}) \]

7. **Identify length and breadth**:
- Therefore, the possible expressions for the length and breadth are:
\[ \text{Length} = (x + 3\sqrt{5}), \quad \text{Breadth} = (x + 2\sqrt{5}) \]

### Part (ii): Area = \(24x^2 - 26x - 8\) sq. units

1. **Identify the area expression**:
The area of the rectangle is given as \(A = 24x^2 - 26x - 8\).

2. **Factor out the common factor**:
- Notice that all terms are divisible by \(2\):
\[ 2(12x^2 - 13x - 4) \]

3. **Factor the quadratic expression**:
- Now we need to factor \(12x^2 - 13x - 4\).
- Here, \(a = 12\), \(b = -13\), and \(c = -4\).
- We need two numbers that multiply to \(ac = 12 \times -4 = -48\) and add to \(-13\).
- The numbers \(-16\) and \(3\) satisfy this condition since:
- \(-16 + 3 = -13\)
- \(-16 \times 3 = -48\)

4. **Rewrite the expression**:
- Rewrite the quadratic as:
\[ 12x^2 - 16x + 3x - 4 \]

5. **Group the terms**:
- Group the first two and the last two terms:
\[ (12x^2 - 16x) + (3x - 4) \]

6. **Factor by grouping**:
- Factor out the common terms:
\[ 4x(3x - 4) + 1(3x - 4) \]
- This gives:
\[ (3x - 4)(4x + 1) \]

7. **Identify length and breadth**:
- Therefore, the possible expressions for the length and breadth are:
\[ \text{Length} = (3x - 4), \quad \text{Breadth} = (4x + 1) \]

### Summary of Solutions:
- For Part (i): Length = \(x + 3\sqrt{5}\), Breadth = \(x + 2\sqrt{5}\)
- For Part (ii): Length = \(3x - 4\), Breadth = \(4x + 1\)