What should be subtracted from `p(x)=6x^(4)+7x^(3)+26x^(2)-25x+25` so that the resulting polynomial is exactly divisible by `g(x)=3x^(2)-x+4`.

Answer

To solve the problem of what should be subtracted from the polynomial \( p(x) = 6x^4 + 7x^3 + 26x^2 - 25x + 25 \) so that it becomes exactly divisible by \( g(x) = 3x^2 - x + 4 \), we will perform polynomial long division and find the remainder. The remainder will then be subtracted from \( p(x) \) to make it divisible by \( g(x) \).

### Step-by-Step Solution:

1. **Set Up the Division**: We need to divide \( p(x) \) by \( g(x) \).

\[ p(x) = 6x^4 + 7x^3 + 26x^2 - 25x + 25 \]
\[ g(x) = 3x^2 - x + 4 \]

2. **Divide the Leading Terms**: Divide the leading term of \( p(x) \) by the leading term of \( g(x) \):
\[ \frac{6x^4}{3x^2} = 2x^2 \]

3. **Multiply and Subtract**: Multiply \( g(x) \) by \( 2x^2 \) and subtract from \( p(x) \):
\[ 2x^2 \cdot (3x^2 - x + 4) = 6x^4 - 2x^3 + 8x^2 \]
Now, subtract this from \( p(x) \):
\[ (6x^4 + 7x^3 + 26x^2 - 25x + 25) - (6x^4 - 2x^3 + 8x^2) = 9x^3 + 18x^2 - 25x + 25 \]

4. **Repeat the Process**: Now, divide the new polynomial \( 9x^3 + 18x^2 - 25x + 25 \) by \( g(x) \):
\[ \frac{9x^3}{3x^2} = 3x \]
Multiply \( g(x) \) by \( 3x \):
\[ 3x \cdot (3x^2 - x + 4) = 9x^3 - 3x^2 + 12x \]
Subtract this from the current polynomial:
\[ (9x^3 + 18x^2 - 25x + 25) - (9x^3 - 3x^2 + 12x) = 21x^2 - 37x + 25 \]

5. **Continue Division**: Now divide \( 21x^2 - 37x + 25 \) by \( g(x) \):
\[ \frac{21x^2}{3x^2} = 7 \]
Multiply \( g(x) \) by \( 7 \):
\[ 7 \cdot (3x^2 - x + 4) = 21x^2 - 7x + 28 \]
Subtract this from the current polynomial:
\[ (21x^2 - 37x + 25) - (21x^2 - 7x + 28) = -30x - 3 \]

6. **Find the Remainder**: The remainder after the division is:
\[ R(x) = -30x - 3 \]

7. **Determine What to Subtract**: To make \( p(x) \) exactly divisible by \( g(x) \), we need to subtract the remainder:
\[ \text{What to subtract} = -(-30x - 3) = 30x + 3 \]

### Final Answer:
Thus, the expression that should be subtracted from \( p(x) \) is:
\[ 30x + 3 \]