Factorise:
`x ^(2) + 15x + 56`

Answer

To factorise the expression \( x^2 + 15x + 56 \), we can follow these steps:

### Step 1: Identify the coefficients
The expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 15 \) (coefficient of \( x \))
- \( c = 56 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that:
- Multiply to \( ac = 1 \times 56 = 56 \)
- Add up to \( b = 15 \)

After checking the pairs of factors of 56, we find:
- \( 8 \) and \( 7 \) satisfy both conditions:
- \( 8 \times 7 = 56 \)
- \( 8 + 7 = 15 \)

### Step 3: Rewrite the middle term
We can rewrite the expression \( x^2 + 15x + 56 \) by splitting the middle term using the numbers we found:
\[ x^2 + 8x + 7x + 56 \]

### Step 4: Group the terms
Now, we will group the terms:
\[ (x^2 + 8x) + (7x + 56) \]

### Step 5: Factor out the common terms
Now, we factor out the common factors from each group:
1. From \( x^2 + 8x \), we can factor out \( x \):
\[ x(x + 8) \]
2. From \( 7x + 56 \), we can factor out \( 7 \):
\[ 7(x + 8) \]

### Step 6: Combine the factors
Now, we can combine the factored terms:
\[ x(x + 8) + 7(x + 8) \]
This can be factored further:
\[ (x + 8)(x + 7) \]

### Final Answer
Thus, the factorised form of \( x^2 + 15x + 56 \) is:
\[ (x + 8)(x + 7) \]