If Q is the quotient when `(x^(2)-10x-24)` is divided by `(x+2) and xne-2`, which of the following represents Q?

Options

A. `x-22`
B. `x-12`
C. `x+12`
D. `x+22`

Explanation

To find the quotient \( Q \) when \( x^2 - 10x - 24 \) is divided by \( x + 2 \), we can use either factorization or long division. Here, we will go through both methods step by step.

### Method 1: Factorization

1. **Write the expression**: We start with the polynomial \( x^2 - 10x - 24 \).

2. **Factor the quadratic**: We need to express \( x^2 - 10x - 24 \) in factored form. We look for two numbers that multiply to \(-24\) (the constant term) and add up to \(-10\) (the coefficient of \( x \)).
- The numbers that satisfy this condition are \(-12\) and \(2\).
- Thus, we can rewrite the expression as:
\[ x^2 - 12x + 2x - 24 \]

3. **Group the terms**: We can group the terms to factor by grouping:
\[ (x^2 - 12x) + (2x - 24) \]

4. **Factor out common terms**:
- From the first group \( x^2 - 12x \), we can factor out \( x \):
\[ x(x - 12) \]
- From the second group \( 2x - 24 \), we can factor out \( 2 \):
\[ 2(x - 12) \]

5. **Combine the factors**: Now we can combine the factored terms:
\[ x(x - 12) + 2(x - 12) = (x + 2)(x - 12) \]

6. **Divide by \( x + 2 \)**: Since we want to find \( Q \), we divide the factored expression by \( x + 2 \):
\[ Q = \frac{(x + 2)(x - 12)}{(x + 2)} \]

7. **Cancel out the common factor**: The \( x + 2 \) terms cancel out (since \( x \neq -2 \)):
\[ Q = x - 12 \]

### Conclusion:
The quotient \( Q \) is:
\[ Q = x - 12 \]

### Method 2: Long Division

1. **Set up the long division**: We will divide \( x^2 - 10x - 24 \) by \( x + 2 \).

2. **Divide the leading terms**: Divide the leading term of the dividend \( x^2 \) by the leading term of the divisor \( x \):
\[ x^2 \div x = x \]

3. **Multiply and subtract**: Multiply \( x \) by \( x + 2 \) and subtract from the original polynomial:
\[ x(x + 2) = x^2 + 2x \]
\[ (x^2 - 10x - 24) - (x^2 + 2x) = -12x - 24 \]

4. **Repeat the process**: Now, divide the leading term \(-12x\) by \(x\):
\[ -12x \div x = -12 \]
Multiply and subtract again:
\[ -12(x + 2) = -12x - 24 \]
\[ (-12x - 24) - (-12x - 24) = 0 \]

5. **Conclude the division**: Since there is no remainder, the quotient is:
\[ Q = x - 12 \]

### Final Answer:
The quotient \( Q \) when \( x^2 - 10x - 24 \) is divided by \( x + 2 \) is:
\[ Q = x - 12 \]