Find the number of possible common
tangents that exist for the following pairs
of circles.
`x^(2) + y^(2) + 6x + 6y + 14 = 0`
`x^(2) + y(2) - 2x -4y -4 =0`

Answer

To find the number of possible common tangents for the given pairs of circles, we will first rewrite the equations of the circles in standard form and then analyze their positions relative to each other.

### Step 1: Rewrite the equations of the circles

The equations of the circles are given as:

1. \( x^2 + y^2 + 6x + 6y + 14 = 0 \)
2. \( x^2 + y^2 - 2x - 4y - 4 = 0 \)

We will rewrite these equations in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \).

#### Circle 1:
Starting with the first circle:
\[ x^2 + y^2 + 6x + 6y + 14 = 0 \]
We can complete the square for \(x\) and \(y\).

- For \(x^2 + 6x\):
\[ x^2 + 6x = (x + 3)^2 - 9 \]

- For \(y^2 + 6y\):
\[ y^2 + 6y = (y + 3)^2 - 9 \]

Substituting back, we have:
\[ (x + 3)^2 - 9 + (y + 3)^2 - 9 + 14 = 0 \]
\[ (x + 3)^2 + (y + 3)^2 - 4 = 0 \]
\[ (x + 3)^2 + (y + 3)^2 = 4 \]

This represents a circle with center \((-3, -3)\) and radius \(r_1 = 2\).

#### Circle 2:
Now for the second circle:
\[ x^2 + y^2 - 2x - 4y - 4 = 0 \]
Completing the square for \(x\) and \(y\):

- For \(x^2 - 2x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]

- For \(y^2 - 4y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]

Substituting back, we have:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 - 4 = 0 \]
\[ (x - 1)^2 + (y - 2)^2 - 9 = 0 \]
\[ (x - 1)^2 + (y - 2)^2 = 9 \]

This represents a circle with center \((1, 2)\) and radius \(r_2 = 3\).

### Step 2: Determine the distance between the centers

Now we need to find the distance \(d\) between the centers of the two circles:
- Center of Circle 1: \((-3, -3)\)
- Center of Circle 2: \((1, 2)\)

Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(1 - (-3))^2 + (2 - (-3))^2} \]
\[ = \sqrt{(1 + 3)^2 + (2 + 3)^2} \]
\[ = \sqrt{4^2 + 5^2} \]
\[ = \sqrt{16 + 25} \]
\[ = \sqrt{41} \]

### Step 3: Analyze the relationship between the circles

Now we compare the distance \(d\) with the sum and difference of the radii:
- \(r_1 + r_2 = 2 + 3 = 5\)
- \(r_2 - r_1 = 3 - 2 = 1\)

### Step 4: Determine the number of common tangents

The number of common tangents can be determined by the following conditions:
1. If \(d > r_1 + r_2\): 4 common tangents
2. If \(d = r_1 + r_2\): 3 common tangents (externally tangent)
3. If \(r_2 - r_1 < d < r_1 + r_2\): 2 common tangents
4. If \(d = r_2 - r_1\): 1 common tangent (internally tangent)
5. If \(d < r_2 - r_1\): 0 common tangents

In our case:
- \(d = \sqrt{41} \approx 6.4\)
- \(r_1 + r_2 = 5\)
- \(d > r_1 + r_2\)

Thus, there are **4 common tangents**.

### Final Answer:
The number of possible common tangents that exist for the given pairs of circles is **4**.