The expression \( x^2 + 2x - y^2 + 1 \) can be analyzed or manipulated in various ways depending on what you need to do with it. Here are a few common approaches:

### Completing the Square
To complete the square for the \( x \)-terms:

1. Start with the \( x \)-terms: \( x^2 + 2x \).
2. Take half of the coefficient of \( x \) (which is 2), square it, and add and subtract this value inside the expression. Half of 2 is 1, and \( 1^2 = 1 \).

So,
\[ x^2 + 2x = (x^2 + 2x + 1) - 1 = (x + 1)^2 - 1. \]

Now, substitute this back into the original expression:
\[ x^2 + 2x - y^2 + 1 = (x + 1)^2 - 1 - y^2 + 1. \]

Simplify:
\[ (x + 1)^2 - y^2. \]

### Factoring
The expression \( (x + 1)^2 - y^2 \) is a difference of squares, which can be factored as:
\[ (x + 1 + y)(x + 1 - y). \]

So, the factored form of the original expression is:
\[ (x + 1 + y)(x + 1 - y). \]

### Interpretation
This expression represents a hyperbola when set equal to a constant, for example:
\[ x^2 + 2x - y^2 + 1 = 0. \]

In this case, it can be rewritten as:
\[ (x + 1)^2 - y^2 = 0, \]
which simplifies to:
\[ (x + 1 + y)(x + 1 - y) = 0. \]

This represents two intersecting lines:
\[ x + 1 + y = 0 \quad \text{and} \quad x + 1 - y = 0. \]

### Summary
The expression \( x^2 + 2x - y^2 + 1 \) can be rewritten as \( (x + 1 + y)(x + 1 - y) \) by completing the square and factoring. This form can be useful for solving equations, graphing, or further algebraic manipulation.