What should be subtracted from `3x^(3)-8x^(2)+4x - 3,` so that the resulting expression has (x+2) as a factor

Answer

To solve the problem of what should be subtracted from the polynomial \(3x^3 - 8x^2 + 4x - 3\) so that the resulting expression has \((x + 2)\) as a factor, we can follow these steps:

### Step 1: Define the polynomial and the term to be subtracted
Let the polynomial be:
\[ f(x) = 3x^3 - 8x^2 + 4x - 3 \]
Let \(k\) be the term that we need to subtract from \(f(x)\). The new expression will be:
\[ f(x) - k \]

### Step 2: Set up the condition for \((x + 2)\) to be a factor
For \((x + 2)\) to be a factor of \(f(x) - k\), it must hold that:
\[ f(-2) - k = 0 \]
This means that \(f(-2) = k\).

### Step 3: Calculate \(f(-2)\)
Now we will substitute \(-2\) into the polynomial \(f(x)\):
\[ f(-2) = 3(-2)^3 - 8(-2)^2 + 4(-2) - 3 \]
Calculating each term:
- \(3(-2)^3 = 3 \times -8 = -24\)
- \(-8(-2)^2 = -8 \times 4 = -32\)
- \(4(-2) = -8\)
- The constant term is \(-3\)

Now, combine these values:
\[ f(-2) = -24 - 32 - 8 - 3 \]
Calculating this step-by-step:
\[ -24 - 32 = -56 \]
\[ -56 - 8 = -64 \]
\[ -64 - 3 = -67 \]
Thus, we have:
\[ f(-2) = -67 \]

### Step 4: Solve for \(k\)
Since \(f(-2) = k\), we find:
\[ k = -67 \]

### Conclusion
The value that should be subtracted from the polynomial \(3x^3 - 8x^2 + 4x - 3\) so that the resulting expression has \((x + 2)\) as a factor is:
\[ \boxed{-67} \]