Factorise:
`p ^(2) - 4p - 77`

Answer

To factorise the expression \( p^2 - 4p - 77 \), we can follow these steps:

### Step 1: Identify the coefficients
The given expression is \( p^2 - 4p - 77 \). Here, the coefficients are:
- Coefficient of \( p^2 \) (which is \( a \)) = 1
- Coefficient of \( p \) (which is \( b \)) = -4
- Constant term (which is \( c \)) = -77

### Step 2: Multiply \( a \) and \( c \)
We need to multiply the coefficient of \( p^2 \) (which is 1) with the constant term (-77):
\[ a \cdot c = 1 \cdot (-77) = -77 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to -77 and add to -4. Let's list the factor pairs of -77:
- \( 1 \) and \( -77 \)
- \( -1 \) and \( 77 \)
- \( 7 \) and \( -11 \)
- \( -7 \) and \( 11 \)

Among these, the pair \( 7 \) and \( -11 \) satisfies our requirement:
\[ 7 + (-11) = -4 \quad \text{and} \quad 7 \cdot (-11) = -77 \]

### Step 4: Rewrite the expression
We can now rewrite the middle term (-4p) using the numbers we found:
\[ p^2 + 7p - 11p - 77 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (p^2 + 7p) + (-11p - 77) \]

### Step 6: Factor by grouping
Now, we factor out the common factors in each group:
\[ p(p + 7) - 11(p + 7) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (p + 7) \):
\[ (p + 7)(p - 11) \]

### Final Result
Thus, the factorised form of \( p^2 - 4p - 77 \) is:
\[ (p + 7)(p - 11) \]

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