Add: `2x^(3) - 9x^(2) + 8, 3x^(2) - 6x - 5, 7x^(3) - 10x + 1` and `3 + 2x - 5x^(2) - 4x^(3)`

Answer

To solve the problem of adding the algebraic expressions \(2x^3 - 9x^2 + 8\), \(3x^2 - 6x - 5\), \(7x^3 - 10x + 1\), and \(3 + 2x - 5x^2 - 4x^3\), we will follow these steps:

### Step 1: Write down the expressions
We have the following expressions to add:
1. \(2x^3 - 9x^2 + 8\)
2. \(3x^2 - 6x - 5\)
3. \(7x^3 - 10x + 1\)
4. \(3 + 2x - 5x^2 - 4x^3\)

### Step 2: Group like terms
We will group the terms based on their degree (the power of \(x\)):
- **Cubic terms**: \(2x^3\), \(7x^3\), and \(-4x^3\)
- **Quadratic terms**: \(-9x^2\), \(3x^2\), and \(-5x^2\)
- **Linear terms**: \(-6x\), \(-10x\), and \(2x\)
- **Constant terms**: \(8\), \(-5\), and \(3\)

### Step 3: Add the cubic terms
\[ 2x^3 + 7x^3 - 4x^3 = (2 + 7 - 4)x^3 = 5x^3 \]

### Step 4: Add the quadratic terms
\[ -9x^2 + 3x^2 - 5x^2 = (-9 + 3 - 5)x^2 = -11x^2 \]

### Step 5: Add the linear terms
\[ -6x - 10x + 2x = (-6 - 10 + 2)x = -14x \]

### Step 6: Add the constant terms
\[ 8 - 5 + 3 = 6 \]

### Step 7: Combine all the results
Now, we combine all the terms we calculated:
\[ 5x^3 - 11x^2 - 14x + 6 \]

### Final Answer
Thus, the sum of the given algebraic expressions is:
\[ \boxed{5x^3 - 11x^2 - 14x + 6} \]