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 given circles, we will first rewrite the equations of the circles in standard form, identify their centers and radii, and then apply the conditions for the number of common tangents.

### Step 1: Rewrite the equations of the circles in standard form.

The first circle is given by:
\[ x^2 + y^2 - 2x - 4y - 4 = 0 \]

Rearranging it:
\[ (x^2 - 2x) + (y^2 - 4y) = 4 \]
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 \( r_1 = \sqrt{9} = 3 \)

The second circle is given by:
\[ x^2 + y^2 + 4x + 8y - 5 = 0 \]

Rearranging it:
\[ (x^2 + 4x) + (y^2 + 8y) = 5 \]
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 \( r_2 = \sqrt{25} = 5 \)

### Step 2: Calculate the distance between the centers of the circles.

The distance \( d \) between the centers \( C_1(1, 2) \) and \( C_2(-2, -4) \) is calculated as follows:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ d = \sqrt{((-2) - 1)^2 + ((-4) - 2)^2} \]
\[ d = \sqrt{(-3)^2 + (-6)^2} \]
\[ d = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \]

### Step 3: Determine the number of common tangents.

To find the number of common tangents, we use the following conditions:
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

Calculating \( r_1 + r_2 \) and \( |r_1 - r_2| \):
- \( r_1 + r_2 = 3 + 5 = 8 \)
- \( |r_1 - r_2| = |3 - 5| = 2 \)

Now we compare:
- \( d = 3\sqrt{5} \approx 6.71 \)
- \( r_1 + r_2 = 8 \)
- \( |r_1 - r_2| = 2 \)

Since \( 2 < 3\sqrt{5} < 8 \), we have:
\[ |r_1 - r_2| < d < r_1 + r_2 \]

Thus, the circles intersect, and there are **2 common tangents**.

### Final Answer:
The number of common tangents to the circles is **2**.