The expression `x^(2)-x-12` can be written as the product of two binomial factors with integer coefficients. One of the binomials is `(x+3)`. Which of the following is the other binomia ?

Options

A. `x^(2)-4`
B. `x^(2)+4`
C. `x-4`
D. `x+4`

Answer

To solve the problem of factoring the expression \( x^2 - x - 12 \) given that one of the binomials is \( (x + 3) \), we can follow these steps:

### Step 1: Identify the expression to factor
We start with the expression:
\[ x^2 - x - 12 \]

### Step 2: Set up the factorization
Since we know one of the binomials is \( (x + 3) \), we can express the quadratic as:
\[ (x + 3)(x + b) \]
where \( b \) is the unknown that we need to find.

### Step 3: Expand the expression
Now, we expand the expression:
\[ (x + 3)(x + b) = x^2 + bx + 3x + 3b = x^2 + (b + 3)x + 3b \]

### Step 4: Match coefficients
We need to match the coefficients of the expanded expression with those in the original expression \( x^2 - x - 12 \).

1. The coefficient of \( x \) gives us the equation:
\[ b + 3 = -1 \]
Solving for \( b \):
\[ b = -1 - 3 = -4 \]

2. The constant term gives us:
\[ 3b = -12 \]
Substituting \( b = -4 \):
\[ 3(-4) = -12 \]
This confirms our value for \( b \).

### Step 5: Write the other binomial
Now that we have found \( b = -4 \), we can write the other binomial:
\[ x + b = x - 4 \]

### Conclusion
Thus, the expression \( x^2 - x - 12 \) can be factored as:
\[ (x + 3)(x - 4) \]
The other binomial is:
\[ \boxed{x - 4} \]