factorize the given expression `15x^2-x-28`

Explanation

To factorize the expression \( 15x^2 - x - 28 \), we will use the middle term splitting method. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given expression is \( 15x^2 - x - 28 \). Here, the coefficient of \( x^2 \) is \( a = 15 \), the coefficient of \( x \) is \( b = -1 \), and the constant term is \( c = -28 \).

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ a \cdot c = 15 \cdot (-28) = -420 \]

### Step 3: Find two numbers that multiply to \( -420 \) and add to \( -1 \)
We need to find two numbers that multiply to \( -420 \) (the product from Step 2) and add up to \( -1 \) (the coefficient of \( x \)).

The factors of \( -420 \) that satisfy this condition are \( -21 \) and \( 20 \):
\[ -21 + 20 = -1 \quad \text{and} \quad -21 \cdot 20 = -420 \]

### Step 4: Rewrite the middle term
We can rewrite the expression \( 15x^2 - x - 28 \) using the two numbers found:
\[ 15x^2 - 21x + 20x - 28 \]

### Step 5: Group the terms
Now, we will group the terms:
\[ (15x^2 - 21x) + (20x - 28) \]

### Step 6: Factor out the common terms from each group
From the first group \( (15x^2 - 21x) \), we can factor out \( 3x \):
\[ 3x(5x - 7) \]
From the second group \( (20x - 28) \), we can factor out \( 4 \):
\[ 4(5x - 7) \]

### Step 7: Combine the factors
Now we can combine the factored terms:
\[ 3x(5x - 7) + 4(5x - 7) = (5x - 7)(3x + 4) \]

### Final Answer
Thus, the factorization of the expression \( 15x^2 - x - 28 \) is:
\[ (5x - 7)(3x + 4) \]

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