If the sum of the coefficients in the expansion of `(2x+3cx+c^(2)x^(2))^(12)` vanishes, then absolute value of sum of the values of `c` is

Explanation

To solve the problem, we need to find the absolute value of the sum of the values of \( c \) such that the sum of the coefficients in the expansion of \( (2x + 3cx + c^2 x^2)^{12} \) vanishes.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The sum of the coefficients in a polynomial can be found by substituting \( x = 1 \). Therefore, we need to evaluate:
\[ (2(1) + 3c(1) + c^2(1^2))^{12} = (2 + 3c + c^2)^{12} \]
We want this expression to equal zero:
\[ 2 + 3c + c^2 = 0 \]

2. **Rearranging the Equation**:
The equation can be rearranged to:
\[ c^2 + 3c + 2 = 0 \]

3. **Factoring the Quadratic**:
We can factor the quadratic equation:
\[ c^2 + 3c + 2 = (c + 1)(c + 2) = 0 \]

4. **Finding the Roots**:
Setting each factor to zero gives us:
\[ c + 1 = 0 \quad \Rightarrow \quad c = -1 \]
\[ c + 2 = 0 \quad \Rightarrow \quad c = -2 \]

5. **Calculating the Sum of Values of \( c \)**:
Now, we find the sum of the values of \( c \):
\[ -1 + (-2) = -3 \]

6. **Finding the Absolute Value**:
Finally, we need the absolute value of the sum:
\[ | -3 | = 3 \]

### Final Answer:
The absolute value of the sum of the values of \( c \) is \( 3 \).