What should be added to `1+2x-3x^(2)` to get `x^(2)-x-1`?

Answer

To solve the problem, we need to determine what should be added to the expression \(1 + 2x - 3x^2\) in order to obtain the expression \(x^2 - x - 1\).

### Step-by-Step Solution:

**Step 1: Write down the equation.**

We want to find an expression \(A\) such that:
\[ (1 + 2x - 3x^2) + A = x^2 - x - 1 \]

**Step 2: Rearrange the equation to isolate \(A\).**

To find \(A\), we can rearrange the equation:
\[ A = (x^2 - x - 1) - (1 + 2x - 3x^2) \]

**Step 3: Distribute the negative sign.**

Now, we will distribute the negative sign across the expression \(1 + 2x - 3x^2\):
\[ A = x^2 - x - 1 - 1 - 2x + 3x^2 \]

**Step 4: Combine like terms.**

Next, we will combine the like terms:
- For \(x^2\): \(x^2 + 3x^2 = 4x^2\)
- For \(x\): \(-x - 2x = -3x\)
- For the constant terms: \(-1 - 1 = -2\)

Putting it all together, we have:
\[ A = 4x^2 - 3x - 2 \]

### Final Answer:
Thus, the expression that should be added to \(1 + 2x - 3x^2\) to get \(x^2 - x - 1\) is:
\[ 4x^2 - 3x - 2 \]