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The system of equations presented is:

1. \((X - 3)^2 + (Y - 4)^2 = 16\)
2. \(X + Y + 2 = 0\)

### Step 1: Interpret the equations
The first equation represents a **circle** centered at \((3, 4)\) with a radius of \(4\) (since \(\sqrt{16} = 4\)).

The second equation, \(X + Y + 2 = 0\), can be rewritten as \(Y = -X - 2\), which is the equation of a **line**.

### Step 2: Solve the system

We substitute \(Y = -X - 2\) into the circle equation to solve for \(X\).

\[
(X - 3)^2 + (-X - 2 - 4)^2 = 16
\]
Simplify the expression inside the second parenthesis:
\[
(X - 3)^2 + (-X - 6)^2 = 16
\]
Now expand both terms:
\[
(X - 3)^2 = X^2 - 6X + 9
\]
\[
(-X - 6)^2 = X^2 + 12X + 36
\]
Substitute these into the equation:
\[
(X^2 - 6X + 9) + (X^2 + 12X + 36) = 16
\]
Combine like terms:
\[
2X^2 + 6X + 45 = 16
\]
Simplify further:
\[
2X^2 + 6X + 29 = 0
\]
Now, divide by 2 to simplify:
\[
X^2 + 3X + \frac{29}{2} = 0
\]

### Step 3: Discriminant analysis

The quadratic equation \(X^2 + 3X + \frac{29}{2} = 0\) can be solved using the discriminant \(\Delta\), where:

\[
\Delta = b^2 - 4ac
\]

For \(a = 1\), \(b = 3\), and \(c = \frac{29}{2}\):
\[
\Delta = 3^2 - 4(1)\left(\frac{29}{2}\right) = 9 - 58 = -49
\]

### Step 4: Conclusion

Since the discriminant \(\Delta = -49\) is negative, the quadratic equation has no real solutions. This means the line does not intersect the circle at any real point.

### Final Answer:
The system of equations has **no real solutions**.

Would you like any further details or explanations? Here are some related questions you might find helpful:

1. What is the significance of the discriminant in quadratic equations?
2. How can you determine the number of intersection points between a line and a circle?
3. What happens if the discriminant is zero or positive?
4. Can this system of equations have complex solutions?
5. How do you graph a circle and a line to visualize their relationship?

**Tip:** When solving systems involving quadratic and linear equations, always check the discriminant to quickly determine the nature of the solutions.

How many real solutions will the following system of equations have: (X - 3)^2 + (Y - 4)^2 = 16; X + Y + 2 = 0?

This problem involves solving a system of equations where a circle and a line are given. Learn how to analyze the intersection of a circle and a line using the quadratic discriminant to determine that the system has no real solutions. Suitable for high school students.

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