Factorise:
`y ^(2)- 6y - 135`

Answer

To factorise the expression \( y^2 - 6y - 135 \), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the standard quadratic form \( ay^2 + by + c \), where:
- \( a = 1 \) (coefficient of \( y^2 \))
- \( b = -6 \) (coefficient of \( y \))
- \( c = -135 \) (constant term)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ a \cdot c = 1 \cdot (-135) = -135 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We are looking for two numbers that multiply to \(-135\) and add up to \(-6\).

After testing pairs of factors of \(-135\), we find:
- The pair \( 9 \) and \(-15\) works because:
\[ 9 \cdot (-15) = -135 \quad \text{and} \quad 9 + (-15) = -6 \]

### Step 4: Rewrite the middle term
Using the numbers \( 9 \) and \(-15\), we can rewrite the expression:
\[ y^2 - 6y - 135 = y^2 + 9y - 15y - 135 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (y^2 + 9y) + (-15y - 135) \]

### Step 6: Factor by grouping
Now, we factor out the common factors in each group:
1. From the first group \( (y^2 + 9y) \), we can factor out \( y \):
\[ y(y + 9) \]
2. From the second group \( (-15y - 135) \), we can factor out \(-15\):
\[ -15(y + 9) \]

### Step 7: Combine the factors
Now, we can combine the factored groups:
\[ y(y + 9) - 15(y + 9) = (y - 15)(y + 9) \]

### Final Answer
Thus, the factorised form of \( y^2 - 6y - 135 \) is:
\[ (y - 15)(y + 9) \]

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