The expression $x^2 + 1$ is a quadratic expression in terms of $x$. Here's an overview of its properties:

1. Standard Form: This expression is already in its simplest form, $ax^2 + bx + c$, where $a = 1$, $b = 0$, and $c = 1$.

2. Roots: To find the roots of $x^2 + 1 = 0$, we set the expression equal to zero:

   $x^2 + 1 = 0$ $x^2 = -1$

   Since $x^2 = -1$, the solutions for $x$ are imaginary numbers:

   $x = \pm i$ where $i$ is the imaginary unit ($i^2 = -1$).

3. Graph: The graph of $y = x^2 + 1$ is a parabola opening upwards, with its vertex at $(0, 1)$. It does not intersect the x-axis because there are no real roots.

Would you like a detailed explanation of any of these points? Here are a few related questions:

1. How can complex numbers be used in solving quadratic equations?
2. Can we apply the quadratic formula to $x^2 + 1 = 0$?
3. What is the importance of the imaginary unit $i$?
4. How do we graph quadratic equations with no real roots?
5. What is the vertex form of a quadratic equation?

Tip: When dealing with imaginary numbers, remember that they arise when the square of a number is negative.