What must be subtracted from `3x^(3) - 5x^(2) - x+2 " to get " 4x^(3) + 3x - 5 `?

Answer

To find out what must be subtracted from \(3x^3 - 5x^2 - x + 2\) to get \(4x^3 + 3x - 5\), we can set up the equation as follows:

1. **Set up the equation:**
We want to find \(p\) such that:
\[ 3x^3 - 5x^2 - x + 2 - p = 4x^3 + 3x - 5 \]

2. **Rearranging the equation:**
To isolate \(p\), we can rearrange the equation:
\[ p = (3x^3 - 5x^2 - x + 2) - (4x^3 + 3x - 5) \]

3. **Distributing the negative sign:**
When we subtract \(4x^3 + 3x - 5\), we need to distribute the negative sign:
\[ p = 3x^3 - 5x^2 - x + 2 - 4x^3 - 3x + 5 \]

4. **Combining like terms:**
Now, we combine the like terms:
- For \(x^3\) terms: \(3x^3 - 4x^3 = -1x^3\)
- For \(x^2\) terms: \(-5x^2\) (no other \(x^2\) terms to combine with)
- For \(x\) terms: \(-x - 3x = -4x\)
- For constant terms: \(2 + 5 = 7\)

Thus, we have:
\[ p = -x^3 - 5x^2 - 4x + 7 \]

5. **Final answer:**
Therefore, the expression that must be subtracted is:
\[ -x^3 - 5x^2 - 4x + 7 \]