Add the following expressions:
`3x^(3)+2x^(2)-6x+3, 2x^(3)-3x^(2)-x-4, 1+2x-3x^(2)-4x^(3)`

Answer

To solve the problem of adding the following expressions:

1. \(3x^3 + 2x^2 - 6x + 3\)
2. \(2x^3 - 3x^2 - x - 4\)
3. \(1 + 2x - 3x^2 - 4x^3\)

We will follow these steps:

### Step 1: Write down the expressions
First, we will write down the three expressions clearly:

\[ 3x^3 + 2x^2 - 6x + 3 \]
\[ 2x^3 - 3x^2 - x - 4 \]
\[ 1 + 2x - 3x^2 - 4x^3 \]

### Step 2: Combine like terms
Next, we will combine the like terms from all three expressions. We will group the terms by their degrees:

- **Cubic terms** (\(x^3\)): \(3x^3 + 2x^3 - 4x^3\)
- **Quadratic terms** (\(x^2\)): \(2x^2 - 3x^2 - 3x^2\)
- **Linear terms** (\(x\)): \(-6x - x + 2x\)
- **Constant terms**: \(3 - 4 + 1\)

### Step 3: Calculate each group
Now we will calculate each group:

1. **Cubic terms**:
\[ 3x^3 + 2x^3 - 4x^3 = (3 + 2 - 4)x^3 = 1x^3 = x^3 \]

2. **Quadratic terms**:
\[ 2x^2 - 3x^2 - 3x^2 = (2 - 3 - 3)x^2 = -4x^2 \]

3. **Linear terms**:
\[ -6x - x + 2x = (-6 - 1 + 2)x = -5x \]

4. **Constant terms**:
\[ 3 - 4 + 1 = 0 \]

### Step 4: Write the final expression
Now we can combine all the results from the calculations:

\[ x^3 - 4x^2 - 5x + 0 \]

Since adding zero does not change the value, we can simplify this to:

\[ x^3 - 4x^2 - 5x \]

### Final Answer:
The sum of the given expressions is:

\[ \boxed{x^3 - 4x^2 - 5x} \]