Factorise `24 x ^(2) - 54 x - 15`

Options

A. `3 (2x + 5) (4x -1)`
B. `3 (2x - 5) (4x +1)`
C. `2 (3x - 5) (3x +1)`
D. `4 (2x - 5) (3x +1)`

Answer

To factorize the expression \( 24x^2 - 54x - 15 \), we can follow these steps:

### Step 1: Identify the coefficients
The expression is in the form \( ax^2 + bx + c \), where:
- \( a = 24 \)
- \( b = -54 \)
- \( c = -15 \)

### Step 2: Calculate the product \( ac \)
We need to calculate \( ac \):
\[ ac = 24 \times (-15) = -360 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \(-360\) and add to \(-54\). After checking the factors of \(-360\), we find:
- The numbers are \(-60\) and \(6\) because:
\[ -60 + 6 = -54 \quad \text{and} \quad -60 \times 6 = -360 \]

### Step 4: Rewrite the middle term using the two numbers
Now we can rewrite the expression:
\[ 24x^2 - 60x + 6x - 15 \]

### Step 5: Factor by grouping
Group the terms:
\[ (24x^2 - 60x) + (6x - 15) \]
Now, factor out the common factors from each group:
\[ 12x(2x - 5) + 3(2x - 5) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \((2x - 5)\):
\[ (12x + 3)(2x - 5) \]

### Step 7: Simplify the first factor
Notice that \(12x + 3\) can be simplified:
\[ 12x + 3 = 3(4x + 1) \]
Thus, we can rewrite the expression as:
\[ 3(4x + 1)(2x - 5) \]

### Final Answer
The factorized form of \( 24x^2 - 54x - 15 \) is:
\[ 3(4x + 1)(2x - 5) \]

---