Each of the following is a term in the polynomial which is the product of `(x + 1), (3x ^(2) + 6x) and (2x ^(2) + 6x -1)` except.

Options

A. `6x ^(5)`
B. `36x ^(4)`
C. `- 6x`
D. `-1`

Answer

To solve the problem, we need to find the polynomial that results from the product of the three given expressions: \( (x + 1) \), \( (3x^2 + 6x) \), and \( (2x^2 + 6x - 1) \). Then we will identify which term is not a part of this polynomial.

### Step 1: Expand the first two factors
We start by multiplying the first two expressions:

\[ (x + 1)(3x^2 + 6x) \]

Using the distributive property (also known as the FOIL method for binomials), we get:

\[ = x(3x^2) + x(6x) + 1(3x^2) + 1(6x) \]
\[ = 3x^3 + 6x^2 + 3x^2 + 6x \]
\[ = 3x^3 + 9x^2 + 6x \]

### Step 2: Multiply the result with the third factor
Now we multiply the result from Step 1 with the third expression \( (2x^2 + 6x - 1) \):

\[ (3x^3 + 9x^2 + 6x)(2x^2 + 6x - 1) \]

We will distribute each term from the first polynomial to each term in the second polynomial.

1. **Multiply \( 3x^3 \) with each term in \( (2x^2 + 6x - 1) \)**:
\[ 3x^3 \cdot 2x^2 = 6x^5 \]
\[ 3x^3 \cdot 6x = 18x^4 \]
\[ 3x^3 \cdot (-1) = -3x^3 \]

2. **Multiply \( 9x^2 \) with each term in \( (2x^2 + 6x - 1) \)**:
\[ 9x^2 \cdot 2x^2 = 18x^4 \]
\[ 9x^2 \cdot 6x = 54x^3 \]
\[ 9x^2 \cdot (-1) = -9x^2 \]

3. **Multiply \( 6x \) with each term in \( (2x^2 + 6x - 1) \)**:
\[ 6x \cdot 2x^2 = 12x^3 \]
\[ 6x \cdot 6x = 36x^2 \]
\[ 6x \cdot (-1) = -6x \]

### Step 3: Combine all the terms
Now we combine all the terms obtained from the multiplications:

\[ 6x^5 + (18x^4 + 18x^4) + (-3x^3 + 54x^3 + 12x^3) + (-9x^2 + 36x^2) - 6x \]

This simplifies to:

\[ 6x^5 + 36x^4 + 63x^3 + 27x^2 - 6x \]

### Step 4: Identify the terms
The polynomial we obtained is:

\[ 6x^5 + 36x^4 + 63x^3 + 27x^2 - 6x \]

### Conclusion
Now, we need to identify which term is not a part of this polynomial. The terms present are \( 6x^5, 36x^4, 63x^3, 27x^2, -6x \). Any term not matching this list is the answer to the question.