Factors of `(42-x-x^(2))` are

Options

A. `(x-7), (x-6)`
B. `(x+7), (x-6)`
C. `(x+7), (6-x)`
D. `(x-7), (x+6)`

Answer

To factor the expression \(42 - x - x^2\), we will follow these steps:

### Step 1: Rewrite the expression
First, we will rewrite the expression in standard form:
\[ -x^2 - x + 42 \]

### Step 2: Factor out -1
Next, we will factor out \(-1\) from the expression:
\[ -(x^2 + x - 42) \]

### Step 3: Factor the quadratic expression
Now, we need to factor the quadratic expression \(x^2 + x - 42\). We are looking for two numbers that multiply to \(-42\) (the constant term) and add up to \(1\) (the coefficient of \(x\)).

The two numbers that satisfy this condition are \(7\) and \(-6\) because:
\[ 7 \times (-6) = -42 \quad \text{and} \quad 7 + (-6) = 1 \]

### Step 4: Write the factors
Using these numbers, we can write the quadratic expression as:
\[ (x + 7)(x - 6) \]

### Step 5: Include the factored out -1
Now, substituting back the \(-1\) we factored out earlier, we have:
\[ -(x + 7)(x - 6) \]

### Step 6: Final expression
Thus, the factors of the original expression \(42 - x - x^2\) are:
\[ -(x + 7)(x - 6) \]

### Conclusion
The factors of \(42 - x - x^2\) can be expressed as:
\[ (6 - x)(7 + x) \]
or equivalently,
\[ (x + 7)(6 - x) \]