`m^(2)-5m-36` =_______

Options

A. `(m+9)(m-4)`
B. `(m-9)(m-4)`
C. `(m+9)(m+4)`
D. `(m-9)(m+4)`

Answer

To factor the expression \( m^2 - 5m - 36 \), 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 \( m^2 \))
- \( b = -5 \) (coefficient of \( m \))
- \( c = -36 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times -36 = -36 \) and add to \( b = -5 \).

### Step 3: List the factor pairs of -36
The factor pairs of -36 are:
- \( 1 \) and \( -36 \)
- \( 2 \) and \( -18 \)
- \( 3 \) and \( -12 \)
- \( 4 \) and \( -9 \)
- \( -1 \) and \( 36 \)
- \( -2 \) and \( 18 \)
- \( -3 \) and \( 12 \)
- \( -4 \) and \( 9 \)

### Step 4: Find the correct pair
We need to find a pair that adds up to -5. Checking the pairs:
- \( 4 + (-9) = -5 \) (This is the correct pair)

### Step 5: Rewrite the middle term using the pair
Now we can rewrite the expression \( m^2 - 5m - 36 \) as:
\[ m^2 + 4m - 9m - 36 \]

### Step 6: Factor by grouping
Group the terms:
\[ (m^2 + 4m) + (-9m - 36) \]

Now factor out the common terms:
\[ m(m + 4) - 9(m + 4) \]

### Step 7: Factor out the common binomial
Now we can factor out \( (m + 4) \):
\[ (m + 4)(m - 9) \]

### Final Answer
Thus, the factorization of \( m^2 - 5m - 36 \) is:
\[ (m - 9)(m + 4) \]

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