19. Factor \( x^{3}-5 x^{2}-49 x+245 \) by grouping a \( (x-5)(x-7)^{2} \) b \( (x+5)(x-7)^{2} \) c \( (x+5)(x+7)(x-7) \) d \( (x-5)(x+7)(x-7) \)

To factor the polynomial \( x^{3}-5 x^{2}-49 x+245 \) by grouping, we can follow these steps: 1. **Group the terms**: We can group the polynomial into two parts: \[ (x^3 - 5x^2) + (-49x + 245) \] 2. **Factor out the common factors in each group**: - From the first group \( x^3 - 5x^2 \), we can factor out \( x^2 \): \[ x^2(x - 5) \] - From the second group \( -49x + 245 \), we can factor out \(-49\): \[ -49(x - 5) \] 3. **Combine the factored groups**: Now we can write the polynomial as: \[ x^2(x - 5) - 49(x - 5) \] We can factor out the common factor \( (x - 5) \): \[ (x - 5)(x^2 - 49) \] 4. **Factor the difference of squares**: The expression \( x^2 - 49 \) can be factored as: \[ (x - 7)(x + 7) \] 5. **Combine all factors**: Thus, we have: \[ (x - 5)(x - 7)(x + 7) \] Now, we can check the answer against the provided options: - a \( (x-5)(x-7)^{2} \) - b \( (x+5)(x-7)^{2} \) - c \( (x+5)(x+7)(x-7) \) - d \( (x-5)(x+7)(x-7) \) The correct factorization we found is: \[ (x - 5)(x - 7)(x + 7) \] This matches option **d** \( (x-5)(x+7)(x-7) \). Thus, the answer is: \[ \boxed{d} \]