Factorise:
`x ^(2) + 5x - 104`

Answer

To factorise the expression \( x^2 + 5x - 104 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is:
\[ x^2 + 5x - 104 \]
Here, the coefficient of \( x^2 \) (let's call it \( a \)) is 1, the coefficient of \( x \) (let's call it \( b \)) is 5, and the constant term (let's call it \( c \)) is -104.

### Step 2: Multiply \( a \) and \( c \)
Next, we multiply \( a \) and \( c \):
\[ a \cdot c = 1 \cdot (-104) = -104 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to -104 and add up to 5. The numbers that satisfy this condition are 13 and -8, because:
\[ 13 \cdot (-8) = -104 \quad \text{and} \quad 13 + (-8) = 5 \]

### Step 4: Rewrite the middle term
Using the two numbers we found, we can rewrite the expression:
\[ x^2 + 13x - 8x - 104 \]

### Step 5: Group the terms
Now, we will group the terms:
\[ (x^2 + 13x) + (-8x - 104) \]

### Step 6: Factor out the common terms
Now, we factor out the common terms from each group:
\[ x(x + 13) - 8(x + 13) \]

### Step 7: Factor by grouping
Now, we can factor out \( (x + 13) \):
\[ (x + 13)(x - 8) \]

### Final Answer
Thus, the factorised form of the expression \( x^2 + 5x - 104 \) is:
\[ (x - 8)(x + 13) \]

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