If a*b means a is added to b, and if `a= x^(2)+x+1" and "b= 2x^(2)-4x+6y`, find a * 3b.

Options

A. `11x^(2)-7y+19x-1`
B. `7x^(2)-11x+7y+18`
C. `18x^(2)-7x+11y+1`
D. `7x^(2)-11x+18y+1`

Answer

To solve the problem step by step, we need to find the value of \( a * 3b \) where \( a = x^2 + x + 1 \) and \( b = 2x^2 - 4x + 6y \).

### Step 1: Calculate \( 3b \)
First, we need to calculate \( 3b \):
\[ b = 2x^2 - 4x + 6y \]
Now, multiply \( b \) by 3:
\[ 3b = 3 \times (2x^2 - 4x + 6y) \]
Distributing the 3:
\[ 3b = 3 \times 2x^2 - 3 \times 4x + 3 \times 6y = 6x^2 - 12x + 18y \]

### Step 2: Add \( a \) and \( 3b \)
Now that we have \( 3b \), we can find \( a * 3b \) which means we need to add \( a \) and \( 3b \):
\[ a = x^2 + x + 1 \]
So, we need to calculate:
\[ a + 3b = (x^2 + x + 1) + (6x^2 - 12x + 18y) \]

### Step 3: Combine like terms
Now, we will combine like terms:
- Combine \( x^2 \) terms: \( x^2 + 6x^2 = 7x^2 \)
- Combine \( x \) terms: \( x - 12x = -11x \)
- The constant term is \( 1 \)
- The \( y \) term is \( 18y \)

Putting it all together:
\[ a + 3b = 7x^2 - 11x + 18y + 1 \]

### Final Answer
Thus, the value of \( a * 3b \) is:
\[ \boxed{7x^2 - 11x + 18y + 1} \]