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 solve the problem of what must be added to the polynomial \( P(t) = 11t^3 + 5t^4 + 6t^5 - 3t^2 + t + 5 \) so that it becomes exactly divisible by \( D(t) = 4 - 2t + 3t^2 \), we will follow these steps:

### Step 1: Arrange the Polynomials
First, we need to arrange the polynomial \( P(t) \) in descending order of the powers of \( t \):
\[ P(t) = 6t^5 + 5t^4 + 11t^3 - 3t^2 + t + 5 \]
And the divisor \( D(t) \) is already arranged as:
\[ D(t) = 3t^2 - 2t + 4 \]

### Step 2: Perform Polynomial Long Division
Next, we will perform polynomial long division of \( P(t) \) by \( D(t) \).

1. **Divide the leading term**: Divide \( 6t^5 \) by \( 3t^2 \) to get \( 2t^3 \).
2. **Multiply and subtract**:
\[ 2t^3 \cdot (3t^2 - 2t + 4) = 6t^5 - 4t^4 + 8t^3 \]
Subtract this from \( P(t) \):
\[ (6t^5 + 5t^4 + 11t^3) - (6t^5 - 4t^4 + 8t^3) = 9t^4 + 3t^3 \]
Now, bring down the next term from \( P(t) \):
\[ 9t^4 + 3t^3 - 3t^2 + t + 5 \]

3. **Repeat the process**: Divide \( 9t^4 \) by \( 3t^2 \) to get \( 3t^2 \).
4. **Multiply and subtract**:
\[ 3t^2 \cdot (3t^2 - 2t + 4) = 9t^4 - 6t^3 + 12t^2 \]
Subtract this from the polynomial:
\[ (9t^4 + 3t^3 - 3t^2) - (9t^4 - 6t^3 + 12t^2) = 9t^3 - 15t^2 + t + 5 \]

5. **Continue**: Divide \( 9t^3 \) by \( 3t^2 \) to get \( 3t \).
6. **Multiply and subtract**:
\[ 3t \cdot (3t^2 - 2t + 4) = 9t^3 - 6t^2 + 12t \]
Subtract:
\[ (9t^3 - 15t^2 + t) - (9t^3 - 6t^2 + 12t) = -9t^2 - 11t + 5 \]

7. **Final step**: Divide \( -9t^2 \) by \( 3t^2 \) to get \( -3 \).
8. **Multiply and subtract**:
\[ -3 \cdot (3t^2 - 2t + 4) = -9t^2 + 6t - 12 \]
Subtract:
\[ (-9t^2 - 11t + 5) - (-9t^2 + 6t - 12) = -17t + 17 \]

### Step 3: Find the Remainder
The remainder \( R(t) \) from the division is:
\[ R(t) = -17t + 17 \]

### Step 4: Determine What to Add
To make \( P(t) \) exactly divisible by \( D(t) \), we need to add the negative of the remainder:
\[ \text{What to add} = -R(t) = 17t - 17 \]

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