Factorise:
` y ^(2) + 10y +24`

Answer

To factorise the expression \( y^2 + 10y + 24 \), we can follow these steps:

### Step 1: Identify the coefficients
The given expression is \( y^2 + 10y + 24 \).
- Coefficient of \( y^2 \) (a) = 1
- Coefficient of \( y \) (b) = 10
- Constant term (c) = 24

### Step 2: Multiply \( a \) and \( c \)
We need to multiply the coefficient of \( y^2 \) (which is 1) by the constant term (which is 24):
\[ 1 \times 24 = 24 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to 24 and add up to 10. The pairs of factors of 24 are:
- \( 1 \times 24 \)
- \( 2 \times 12 \)
- \( 3 \times 8 \)
- \( 4 \times 6 \)

Among these, the pair \( 4 \) and \( 6 \) adds up to \( 10 \) (i.e., \( 4 + 6 = 10 \)).

### Step 4: Rewrite the middle term using the two numbers found
We can rewrite the expression \( y^2 + 10y + 24 \) as:
\[ y^2 + 4y + 6y + 24 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (y^2 + 4y) + (6y + 24) \]

Now, we factor out the common factors from each group:
- From the first group \( (y^2 + 4y) \), we can factor out \( y \):
\[ y(y + 4) \]
- From the second group \( (6y + 24) \), we can factor out \( 6 \):
\[ 6(y + 4) \]

### Step 6: Combine the factored terms
Now we have:
\[ y(y + 4) + 6(y + 4) \]
We can see that \( (y + 4) \) is a common factor:
\[ (y + 4)(y + 6) \]

### Final Result
Thus, the factorised form of \( y^2 + 10y + 24 \) is:
\[ (y + 4)(y + 6) \]

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