The equation `x^(2)+y^(2)+2x-10y+30=0` represents the equation of a circle.

Answer

To determine whether the equation \( x^2 + y^2 + 2x - 10y + 30 = 0 \) represents a circle, we will rewrite the equation in standard form and analyze its components.

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x^2 + y^2 + 2x - 10y + 30 = 0 \]

2. **Rearrange the equation:**
Move the constant term to the other side:
\[ x^2 + y^2 + 2x - 10y = -30 \]

3. **Complete the square for the \( x \) terms:**
Take the \( x \) terms \( x^2 + 2x \):
- The coefficient of \( x \) is \( 2 \).
- Half of \( 2 \) is \( 1 \), and squaring it gives \( 1 \).
- Add and subtract \( 1 \):
\[ (x^2 + 2x + 1 - 1) = (x + 1)^2 - 1 \]

4. **Complete the square for the \( y \) terms:**
Take the \( y \) terms \( y^2 - 10y \):
- The coefficient of \( y \) is \( -10 \).
- Half of \( -10 \) is \( -5 \), and squaring it gives \( 25 \).
- Add and subtract \( 25 \):
\[ (y^2 - 10y + 25 - 25) = (y - 5)^2 - 25 \]

5. **Substitute back into the equation:**
Now substitute the completed squares back into the equation:
\[ (x + 1)^2 - 1 + (y - 5)^2 - 25 = -30 \]

6. **Simplify the equation:**
Combine the constants:
\[ (x + 1)^2 + (y - 5)^2 - 26 = -30 \]
\[ (x + 1)^2 + (y - 5)^2 = -30 + 26 \]
\[ (x + 1)^2 + (y - 5)^2 = -4 \]

7. **Analyze the result:**
The equation \( (x + 1)^2 + (y - 5)^2 = -4 \) indicates that the sum of squares equals a negative number, which is not possible. Therefore, this implies that there is no real circle represented by this equation.

### Conclusion:
The given equation does not represent a circle because the radius squared (which would be the right side of the equation after completing the square) is negative.