Factorise :
`6a^(2) - a - 15`

Answer

To factorise the expression \(6a^2 - a - 15\), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is \(6a^2 - a - 15\). Here, the coefficients are:
- \(A = 6\) (coefficient of \(a^2\))
- \(B = -1\) (coefficient of \(a\))
- \(C = -15\) (constant term)

### Step 2: Calculate the product of \(A\) and \(C\)
We need to find the product \(A \times C\):
\[ A \times C = 6 \times (-15) = -90 \]

### Step 3: Find two numbers that multiply to \(A \times C\) and add to \(B\)
We need to find two numbers \(x\) and \(y\) such that:
- \(x \times y = -90\)
- \(x + y = -1\)

After testing possible pairs, we find:
- \(x = 9\) and \(y = -10\) satisfy both conditions:
- \(9 \times (-10) = -90\)
- \(9 + (-10) = -1\)

### Step 4: Rewrite the middle term
Now, we can rewrite the expression \(6a^2 - a - 15\) by splitting the middle term using \(9\) and \(-10\):
\[ 6a^2 + 9a - 10a - 15 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (6a^2 + 9a) + (-10a - 15) \]

### Step 6: Factor by grouping
Now, we factor out the common factors from each group:
- From the first group \(6a^2 + 9a\), we can factor out \(3a\):
\[ 3a(2a + 3) \]
- From the second group \(-10a - 15\), we can factor out \(-5\):
\[ -5(2a + 3) \]

### Step 7: Combine the factors
Now we have:
\[ 3a(2a + 3) - 5(2a + 3) \]
We can factor out the common binomial factor \((2a + 3)\):
\[ (2a + 3)(3a - 5) \]

### Final Answer
Thus, the factorised form of \(6a^2 - a - 15\) is:
\[ (2a + 3)(3a - 5) \]

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