Factorise:
`y ^(2) + 19y + 60`

Answer

To factorise the expression \( y^2 + 19y + 60 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is \( y^2 + 19y + 60 \). Here, the coefficients are:
- Coefficient of \( y^2 \) (a) = 1
- Coefficient of \( y \) (b) = 19
- Constant term (c) = 60

### Step 2: Multiply the coefficient of \( y^2 \) with the constant term
We need to multiply the coefficient of \( y^2 \) (which is 1) with the constant term (which is 60):
\[ 1 \times 60 = 60 \]

### Step 3: Find two numbers that multiply to 60 and add to 19
We need to find two numbers that multiply to 60 and add up to 19. The pairs of factors of 60 are:
- \( 1 \times 60 \)
- \( 2 \times 30 \)
- \( 3 \times 20 \)
- \( 4 \times 15 \)
- \( 5 \times 12 \)
- \( 6 \times 10 \)

Among these pairs, \( 15 \) and \( 4 \) satisfy our conditions:
- \( 15 + 4 = 19 \)
- \( 15 \times 4 = 60 \)

### Step 4: Rewrite the middle term using the two numbers
Now we can rewrite the expression \( y^2 + 19y + 60 \) by splitting the middle term:
\[ y^2 + 15y + 4y + 60 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (y^2 + 15y) + (4y + 60) \]

### Step 6: Factor out the common terms in each group
Now, we factor out the common terms from each group:
- From \( y^2 + 15y \), we can factor out \( y \):
\[ y(y + 15) \]
- From \( 4y + 60 \), we can factor out \( 4 \):
\[ 4(y + 15) \]

### Step 7: Combine the factored terms
Now we can combine the factored terms:
\[ y(y + 15) + 4(y + 15) \]
This gives us:
\[ (y + 15)(y + 4) \]

### Final Result
Thus, the factorization of \( y^2 + 19y + 60 \) is:
\[ (y + 15)(y + 4) \]

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