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

Answer

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

### Step 1: Identify the coefficients
The given quadratic 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 \( b = 15 \)

### Step 3: List the factor pairs of 56
The factor pairs of 56 are:
- \( 1 \times 56 \)
- \( 2 \times 28 \)
- \( 4 \times 14 \)
- \( 7 \times 8 \)

### Step 4: Identify the correct pair
Among these pairs, we look for the pair that adds up to 15. The pair \( 7 \) and \( 8 \) satisfies this condition:
- \( 7 + 8 = 15 \)
- \( 7 \times 8 = 56 \)

### Step 5: Rewrite the middle term
We can rewrite the expression by splitting the middle term \( 15x \) into \( 7x + 8x \):
\[ x^2 + 7x + 8x + 56 \]

### Step 6: Group the terms
Now, we can group the terms:
\[ (x^2 + 7x) + (8x + 56) \]

### Step 7: Factor out the common factors
From the first group \( (x^2 + 7x) \), we can factor out \( x \):
\[ x(x + 7) \]
From the second group \( (8x + 56) \), we can factor out \( 8 \):
\[ 8(x + 7) \]

### Step 8: Combine the factors
Now we can combine the factored terms:
\[ x(x + 7) + 8(x + 7) = (x + 7)(x + 8) \]

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