The number of common tangents to the circle `x^(2)+y^(2)-2x-4y-4=0 and x^(2)+y^(2)+4x+8y-5=0` is _________.

Explanation

To find the number of common tangents to the two given circles, we will follow these steps:

### Step 1: Write the equations of the circles in standard form.
The equations of the circles are given as:
1. \( x^2 + y^2 - 2x - 4y - 4 = 0 \)
2. \( x^2 + y^2 + 4x + 8y - 5 = 0 \)

We will complete the square for both circles.

**For the first circle:**
\[ x^2 - 2x + y^2 - 4y = 4 \]
Completing the square:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 = 4 \]
\[ (x - 1)^2 + (y - 2)^2 = 9 \]
This gives us:
- Center \( C_1 = (1, 2) \)
- Radius \( r_1 = 3 \)

**For the second circle:**
\[ x^2 + 4x + y^2 + 8y = 5 \]
Completing the square:
\[ (x + 2)^2 - 4 + (y + 4)^2 - 16 = 5 \]
\[ (x + 2)^2 + (y + 4)^2 = 25 \]
This gives us:
- Center \( C_2 = (-2, -4) \)
- Radius \( r_2 = 5 \)

### Step 2: Calculate the required values.
Now we will calculate the following values:
1. \( |r_1 - r_2| \)
2. \( r_1 + r_2 \)
3. The distance between the centers \( C_1 \) and \( C_2 \).

**Calculating \( |r_1 - r_2| \):**
\[ |r_1 - r_2| = |3 - 5| = 2 \]

**Calculating \( r_1 + r_2 \):**
\[ r_1 + r_2 = 3 + 5 = 8 \]

**Calculating the distance between the centers \( C_1 \) and \( C_2 \):**
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the values:
\[ d = \sqrt{((-2) - 1)^2 + ((-4) - 2)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} \]

### Step 3: Analyze the conditions for common tangents.
We have:
- \( |r_1 - r_2| = 2 \)
- \( r_1 + r_2 = 8 \)
- \( d = \sqrt{45} \)

Now we need to check the conditions for the number of common tangents:
1. If \( d > r_1 + r_2 \): 4 common tangents
2. If \( d = r_1 + r_2 \): 3 common tangents
3. If \( |r_1 - r_2| < d < r_1 + r_2 \): 2 common tangents
4. If \( d = |r_1 - r_2| \): 1 common tangent
5. If \( d < |r_1 - r_2| \): 0 common tangents

### Step 4: Compare the values.
We know:
- \( |r_1 - r_2| = 2 \)
- \( r_1 + r_2 = 8 \)
- \( d = \sqrt{45} \approx 6.71 \)

Now we check:
- \( 2 < \sqrt{45} < 8 \)

This means that the distance between the centers is greater than the difference of the radii and less than the sum of the radii.

### Conclusion:
Thus, the number of common tangents is **2**.