What must be added to `11t^(3)+5t^(4)+6t^(5)-3t^(2)+t+5`, so that the resulting polynomial is exactly divisible by `4-2t+3t^(2)`?

Explanation

To find what must be added to the polynomial \( P(t) = 11t^3 + 5t^4 + 6t^5 - 3t^2 + t + 5 \) so that it is divisible by \( D(t) = 4 - 2t + 3t^2 \), we will perform polynomial long division and find the remainder. The value we need to add will be the negative of this remainder.

### Step-by-Step Solution:

1. **Arrange the Polynomials**:
We will first write the polynomials in standard form (decreasing powers of \( t \)):
\[ P(t) = 6t^5 + 5t^4 + 11t^3 - 3t^2 + t + 5 \]
\[ D(t) = 3t^2 - 2t + 4 \]

2. **Perform Polynomial Long Division**:
We divide \( P(t) \) by \( D(t) \).

- Divide the leading term of \( P(t) \) (which is \( 6t^5 \)) by the leading term of \( D(t) \) (which is \( 3t^2 \)):
\[ \frac{6t^5}{3t^2} = 2t^3 \]
- Multiply \( D(t) \) by \( 2t^3 \):
\[ 2t^3(3t^2 - 2t + 4) = 6t^5 - 4t^4 + 8t^3 \]
- Subtract this from \( P(t) \):
\[ (6t^5 + 5t^4 + 11t^3) - (6t^5 - 4t^4 + 8t^3) = 9t^4 + 3t^3 \]

3. **Continue the Division**:
- Now, divide \( 9t^4 \) by \( 3t^2 \):
\[ \frac{9t^4}{3t^2} = 3t^2 \]
- Multiply \( D(t) \) by \( 3t^2 \):
\[ 3t^2(3t^2 - 2t + 4) = 9t^4 - 6t^3 + 12t^2 \]
- Subtract:
\[ (9t^4 + 3t^3 - 3t^2) - (9t^4 - 6t^3 + 12t^2) = 9t^3 - 15t^2 \]

4. **Continue the Division**:
- Divide \( 9t^3 \) by \( 3t^2 \):
\[ \frac{9t^3}{3t^2} = 3t \]
- Multiply \( D(t) \) by \( 3t \):
\[ 3t(3t^2 - 2t + 4) = 9t^3 - 6t^2 + 12t \]
- Subtract:
\[ (9t^3 - 15t^2 + t) - (9t^3 - 6t^2 + 12t) = -9t^2 - 11t \]

5. **Continue the Division**:
- Divide \( -9t^2 \) by \( 3t^2 \):
\[ \frac{-9t^2}{3t^2} = -3 \]
- Multiply \( D(t) \) by \( -3 \):
\[ -3(3t^2 - 2t + 4) = -9t^2 + 6t - 12 \]
- Subtract:
\[ (-9t^2 - 11t + 5) - (-9t^2 + 6t - 12) = -17t + 17 \]

6. **Find the Remainder**:
The remainder after the division is:
\[ R(t) = -17t + 17 \]

7. **What to Add**:
To make the polynomial divisible by \( D(t) \), we need to add the opposite of the remainder:
\[ \text{What to add} = -(-17t + 17) = 17t - 17 \]

### Final Answer:
Thus, the expression that must be added to the polynomial \( P(t) \) is:
\[ \boxed{17t - 17} \]