Find the number which should be added to `x^2+ x +3` so that the resulting polynomial is completely divisible by (x + 3).

Answer

To find the number that should be added to the polynomial \( f(x) = x^2 + x + 3 \) so that it becomes completely divisible by \( (x + 3) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Polynomial and the Divisor**:
We have the polynomial \( f(x) = x^2 + x + 3 \) and we want it to be divisible by \( (x + 3) \).

2. **Set Up the Condition for Divisibility**:
For \( f(x) \) to be divisible by \( (x + 3) \), the remainder when \( f(x) \) is divided by \( (x + 3) \) must be zero. According to the Remainder Theorem, we need to find \( f(-3) \).

3. **Substitute \( x = -3 \) into the Polynomial**:
\[ f(-3) = (-3)^2 + (-3) + 3 \]
Calculate each term:
\[ f(-3) = 9 - 3 + 3 \]
Simplifying gives:
\[ f(-3) = 9 \]

4. **Set Up the Equation for the Added Number**:
Let \( k \) be the number we need to add to \( f(x) \). After adding \( k \), the new polynomial becomes:
\[ f(x) + k \]
We need:
\[ f(-3) + k = 0 \]
Substituting \( f(-3) = 9 \):
\[ 9 + k = 0 \]

5. **Solve for \( k \)**:
Rearranging the equation gives:
\[ k = -9 \]

6. **Final Polynomial**:
The new polynomial after adding \( k \) will be:
\[ f(x) + k = x^2 + x + 3 - 9 = x^2 + x - 6 \]

### Conclusion:
The number that should be added to \( x^2 + x + 3 \) to make it completely divisible by \( (x + 3) \) is \( -9 \).