Solving the Equation (x+1)² = 4
This equation presents a unique challenge because we're asked to find a value for x that, when added to 1, squared, results in a negative number. This is impossible within the realm of real numbers. Let's explore why:
Understanding the Problem

Squaring a real number always results in a nonnegative value. Any number, whether positive or negative, when multiplied by itself, always produces a positive or zero result. For example:
 (2)² = 4
 (2)² = 4
 (0)² = 0

Therefore, (x+1)² cannot be equal to 4 if x is a real number.
Introducing Complex Numbers
To solve this equation, we need to delve into the world of complex numbers. Complex numbers are a broader system that extends beyond real numbers. They are expressed in the form a + bi, where:
 a is the real part
 b is the imaginary part
 i is the imaginary unit, defined as the square root of 1 (i² = 1).
Solving the Equation Using Complex Numbers

Take the square root of both sides: √(x+1)² = ±√(4)

Simplify: x + 1 = ±2i

Isolate x: x = 1 ± 2i
Therefore, the solutions to the equation (x+1)² = 4 are x = 1 + 2i and x = 1  2i.
Conclusion
While the equation (x+1)² = 4 has no solutions within the set of real numbers, it does have solutions within the set of complex numbers. This demonstrates the power and necessity of complex numbers in expanding our understanding of mathematical solutions.