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

Answer

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

### Step 1: Identify the coefficients
The given expression is \( a^2 + 10a + 24 \). Here, the coefficients are:
- \( a^2 \) has a coefficient of 1 (which is the coefficient of \( a^2 \)),
- The coefficient of \( a \) is 10,
- The constant term is 24.

### Step 2: Find two numbers that multiply to the constant term and add to the coefficient of \( a \)
We need to find two numbers that:
- Multiply to \( 24 \) (the constant term),
- Add up to \( 10 \) (the coefficient of \( a \)).

The pairs of factors of 24 are:
- \( 1 \times 24 \)
- \( 2 \times 12 \)
- \( 3 \times 8 \)
- \( 4 \times 6 \)

Among these pairs, \( 4 \) and \( 6 \) multiply to \( 24 \) and add up to \( 10 \).

### Step 3: Rewrite the expression using the two numbers
Now we can rewrite the middle term \( 10a \) using \( 4a \) and \( 6a \):
\[ a^2 + 4a + 6a + 24 \]

### Step 4: Group the terms
Next, we group the terms:
\[ (a^2 + 4a) + (6a + 24) \]

### Step 5: Factor out the common factors from each group
Now we factor out the common factors from each group:
- From the first group \( (a^2 + 4a) \), we can factor out \( a \):
\[ a(a + 4) \]
- From the second group \( (6a + 24) \), we can factor out \( 6 \):
\[ 6(a + 4) \]

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

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

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