The factors of
`( 2x ^(3) + 19 x ^(2) + 38 x + 21)` are

Options

A. `( 2x + 3) ( x +5) (x +1)`
B. `( 4x + 9) (2x +1) ( x +1)`
C. `( x +1) (x +4) (2x +1)`
D. `(x +1) (x +7) (2x + 3)`

Answer

To find the factors of the polynomial \( 2x^3 + 19x^2 + 38x + 21 \), we will test the provided options one by one.

### Step-by-Step Solution:

1. **Identify the Options**: The options given are:
- Option 1: \( (2x + 3)(x + 5)(x + 1) \)
- Option 2: \( (4x + 9)(2x + 1)(x + 1) \)
- Option 3: \( (x + 1)(x + 1)(x + 4)(2x + 1) \)
- Option 4: \( (x + 1)(x + 7)(2x + 3) \)

2. **Test Option 1**:
- Compute \( (2x + 3)(x + 5) \):
\[ (2x + 3)(x + 5) = 2x^2 + 10x + 3x + 15 = 2x^2 + 13x + 15 \]
- Now multiply by \( (x + 1) \):
\[ (2x^2 + 13x + 15)(x + 1) = 2x^3 + 2x^2 + 13x^2 + 13x + 15x + 15 = 2x^3 + 15x^2 + 28x + 15 \]
- This does not match \( 2x^3 + 19x^2 + 38x + 21 \). **Eliminate Option 1**.

3. **Test Option 2**:
- Compute \( (4x + 9)(2x + 1) \):
\[ (4x + 9)(2x + 1) = 8x^2 + 4x + 18x + 9 = 8x^2 + 22x + 9 \]
- Now multiply by \( (x + 1) \):
\[ (8x^2 + 22x + 9)(x + 1) = 8x^3 + 8x^2 + 22x^2 + 22x + 9x + 9 = 8x^3 + 30x^2 + 31x + 9 \]
- This does not match \( 2x^3 + 19x^2 + 38x + 21 \). **Eliminate Option 2**.

4. **Test Option 3**:
- Compute \( (x + 1)(x + 1) \):
\[ (x + 1)(x + 1) = x^2 + 2x + 1 \]
- Now multiply by \( (x + 4) \):
\[ (x^2 + 2x + 1)(x + 4) = x^3 + 4x^2 + 2x^2 + 8x + x + 4 = x^3 + 6x^2 + 9x + 4 \]
- Now multiply by \( (2x + 1) \):
\[ (x^3 + 6x^2 + 9x + 4)(2x + 1) = 2x^4 + 12x^3 + 18x^2 + 8x + x^3 + 6x^2 + 9x + 4 \]
- This does not match \( 2x^3 + 19x^2 + 38x + 21 \). **Eliminate Option 3**.

5. **Test Option 4**:
- Compute \( (x + 1)(x + 7) \):
\[ (x + 1)(x + 7) = x^2 + 8x + 7 \]
- Now multiply by \( (2x + 3) \):
\[ (x^2 + 8x + 7)(2x + 3) = 2x^3 + 3x^2 + 16x^2 + 24x + 14x + 21 \]
- Combine like terms:
\[ 2x^3 + (3x^2 + 16x^2) + (24x + 14x) + 21 = 2x^3 + 19x^2 + 38x + 21 \]
- This matches \( 2x^3 + 19x^2 + 38x + 21 \). **Option 4 is correct**.

### Final Answer:
The factors of \( 2x^3 + 19x^2 + 38x + 21 \) are \( (x + 1)(x + 7)(2x + 3) \).