Factorise:
`y ^(2) + y-72`

Answer

To factorise the expression \( y^2 + y - 72 \), we will follow these steps:

### Step 1: Identify the coefficients
The given expression is \( y^2 + y - 72 \).
- The coefficient of \( y^2 \) is 1.
- The constant term is -72.

### Step 2: Multiply the coefficient of \( y^2 \) with the constant term
We multiply the coefficient of \( y^2 \) (which is 1) with the constant term (-72):
\[ 1 \times (-72) = -72 \]

### Step 3: Find two numbers that multiply to -72 and add to 1
We need to find two numbers that:
- Multiply to -72
- Add up to the coefficient of \( y \), which is 1.

The two numbers that satisfy these conditions are 9 and -8:
\[ 9 \times (-8) = -72 \quad \text{and} \quad 9 + (-8) = 1 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the expression \( y^2 + y - 72 \) by splitting the middle term \( y \) into \( 9y - 8y \):
\[ y^2 + 9y - 8y - 72 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (y^2 + 9y) + (-8y - 72) \]

### Step 6: Factor out the common factors from each group
From the first group \( (y^2 + 9y) \), we can factor out \( y \):
\[ y(y + 9) \]
From the second group \( (-8y - 72) \), we can factor out -8:
\[ -8(y + 9) \]

### Step 7: Combine the factors
Now we can combine the two groups:
\[ y(y + 9) - 8(y + 9) \]
This can be factored as:
\[ (y - 8)(y + 9) \]

### Final Answer
Thus, the factorised form of \( y^2 + y - 72 \) is:
\[ (y - 8)(y + 9) \]

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