What must be added to `7x^(3)-3x^(2)+5x+4` in order to get `9x^(3)+x^(2)-x-1`?

Answer

To find out what must be added to the polynomial \(7x^3 - 3x^2 + 5x + 4\) in order to obtain the polynomial \(9x^3 + x^2 - x - 1\), we can set up the equation as follows:

1. **Set up the equation**:
\[ 7x^3 - 3x^2 + 5x + 4 + P(x) = 9x^3 + x^2 - x - 1 \]
Here, \(P(x)\) is the polynomial we need to find.

2. **Rearranging the equation**:
To isolate \(P(x)\), we can subtract \(7x^3 - 3x^2 + 5x + 4\) from both sides:
\[ P(x) = (9x^3 + x^2 - x - 1) - (7x^3 - 3x^2 + 5x + 4) \]

3. **Distributing the negative sign**:
When we subtract the second polynomial, we need to change the signs of each term:
\[ P(x) = 9x^3 + x^2 - x - 1 - 7x^3 + 3x^2 - 5x - 4 \]

4. **Combining like terms**:
Now we combine the like terms:
- For \(x^3\): \(9x^3 - 7x^3 = 2x^3\)
- For \(x^2\): \(x^2 + 3x^2 = 4x^2\)
- For \(x\): \(-x - 5x = -6x\)
- For the constant term: \(-1 - 4 = -5\)

Putting it all together:
\[ P(x) = 2x^3 + 4x^2 - 6x - 5 \]

5. **Final answer**:
Therefore, the polynomial that must be added is:
\[ \boxed{2x^3 + 4x^2 - 6x - 5} \]