How much is `x^(3)-3x^(2)+5x-2` less than `3-2x+x^(2)-x^(3)`?

Answer

To find out how much `x^(3) - 3x^(2) + 5x - 2` is less than `3 - 2x + x^(2) - x^(3)`, we need to subtract the first expression from the second expression.

Let's denote the two expressions as follows:
1. Expression A: `x^(3) - 3x^(2) + 5x - 2`
2. Expression B: `3 - 2x + x^(2) - x^(3)`

We want to calculate:
\[ \text{Difference} = \text{Expression B} - \text{Expression A} \]

### Step 1: Write down the expressions
We have:
\[ \text{Expression B} = 3 - 2x + x^{2} - x^{3} \]
\[ \text{Expression A} = x^{3} - 3x^{2} + 5x - 2 \]

### Step 2: Set up the subtraction
Now we will subtract Expression A from Expression B:
\[ \text{Difference} = (3 - 2x + x^{2} - x^{3}) - (x^{3} - 3x^{2} + 5x - 2) \]

### Step 3: Distribute the negative sign
Distributing the negative sign across Expression A gives us:
\[ \text{Difference} = 3 - 2x + x^{2} - x^{3} - x^{3} + 3x^{2} - 5x + 2 \]

### Step 4: Combine like terms
Now, we will combine the like terms:
- For \(x^{3}\): \(-x^{3} - x^{3} = -2x^{3}\)
- For \(x^{2}\): \(x^{2} + 3x^{2} = 4x^{2}\)
- For \(x\): \(-2x - 5x = -7x\)
- For the constant terms: \(3 + 2 = 5\)

Putting it all together, we have:
\[ \text{Difference} = -2x^{3} + 4x^{2} - 7x + 5 \]

### Final Answer
Thus, `x^(3) - 3x^(2) + 5x - 2` is less than `3 - 2x + x^(2) - x^(3)` by:
\[ -2x^{3} + 4x^{2} - 7x + 5 \]