Divide $44(x^4 - 5x^3 - 24x^2) $ by $ 11x(x-8)$:

Q: Divide $44(x^4 - 5x^3 - 24x^2) $ by $ 11x(x-8)$:

To solve the problem of dividing $44(x^4 - 5x^3 - 24x^2)$ by $11x(x-8)$, we need to perform polynomial division. Let's break down the process step by step to justify why Option 4 is the correct answer. ### Step-by-Step Explanation: 1. **Expression Setup:** - Dividend: $44(x^4 - 5x^3 - 24x^2)$ - Divisor: $11x(x-8)$ 2. **Simplify the Divisor:** - The divisor can be expanded to $11x^2 - 88x$. 3. **Perform the Division:** - First, factor out the common factor from the dividend: \[ 44(x^4 - 5x^3 - 24x^2) = 44x^2(x^2 - 5x - 24) \] - Now divide by the divisor: \[ \frac{44x^2(x^2 - 5x - 24)}{11x(x-8)} \] 4. **Cancel Common Factors:** - Simplify the expression by canceling out common factors: \[ \frac{44x^2}{11x} = 4x \] - This leaves us with: \[ 4x \cdot \frac{x^2 - 5x - 24}{x-8} \] 5. **Factor the Remaining Polynomial:** - Factor $x^2 - 5x - 24$: \[ x^2 - 5x - 24 = (x - 8)(x + 3) \] - Substitute back into the expression: \[ 4x \cdot \frac{(x - 8)(x + 3)}{x-8} \] 6. **Cancel the Common Factor:** - Cancel the $(x-8)$ term: \[ 4x(x + 3) \] ### Conclusion: - The final simplified expression is $4x(x + 3)$, which corresponds to **Option 4**. ### Explanation of Incorrect Options: 1. **Option 1: $4x(x - 8)$** - This option incorrectly suggests that the factor $(x - 8)$ remains in the final expression, which is not the case after cancellation. 2. **Option 2: $4x(x + 8)$** - This option incorrectly uses $(x + 8)$ instead of the correct factor $(x + 3)$. 3. **Option 3: $4x(x - 3)$** - This option incorrectly uses $(x - 3)$ instead of the correct factor $(x + 3)$. By following the logical steps of polynomial division and simplification, we have shown that Option 4, $4x(x + 3)$, is indeed the correct answer.

Question

Accepted Answer

To solve the problem of dividing $44(x^4 - 5x^3 - 24x^2)$ by $11x(x-8)$, we need to perform polynomial division. Let's break down the process step by step to justify why Option 4 is the correct answer.

### Step-by-Step Explanation:

1. **Expression Setup:**
   - Dividend: $44(x^4 - 5x^3 - 24x^2)$
   - Divisor: $11x(x-8)$

2. **Simplify the Divisor:**
   - The divisor can be expanded to $11x^2 - 88x$.

3. **Perform the Division:**
   - First, factor out the common factor from the dividend:
     $$
     44(x^4 - 5x^3 - 24x^2) = 44x^2(x^2 - 5x - 24)
     $$
   - Now divide by the divisor:
     $$
     \frac{44x^2(x^2 - 5x - 24)}{11x(x-8)}
     $$

4. **Cancel Common Factors:**
   - Simplify the expression by canceling out common factors:
     $$
     \frac{44x^2}{11x} = 4x
     $$
   - This leaves us with:
     $$
     4x \cdot \frac{x^2 - 5x - 24}{x-8}
     $$

5. **Factor the Remaining Polynomial:**
   - Factor $x^2 - 5x - 24$:
     $$
     x^2 - 5x - 24 = (x - 8)(x + 3)
     $$
   - Substitute back into the expression:
     $$
     4x \cdot \frac{(x - 8)(x + 3)}{x-8}
     $$

6. **Cancel the Common Factor:**
   - Cancel the $(x-8)$ term:
     $$
     4x(x + 3)
     $$

### Conclusion:

- The final simplified expression is $4x(x + 3)$, which corresponds to **Option 4**.

### Explanation of Incorrect Options:

1. **Option 1: $4x(x - 8)$**
   - This option incorrectly suggests that the factor $(x - 8)$ remains in the final expression, which is not the case after cancellation.

2. **Option 2: $4x(x + 8)$**
   - This option incorrectly uses $(x + 8)$ instead of the correct factor $(x + 3)$.

3. **Option 3: $4x(x - 3)$**
   - This option incorrectly uses $(x - 3)$ instead of the correct factor $(x + 3)$.

By following the logical steps of polynomial division and simplification, we have shown that Option 4, $4x(x + 3)$, is indeed the correct answer.