The expression you've provided is a quadratic equation: \(-x^2 + 2x + 1\).

Let's break it down step-by-step:

### Step 1: Simplify the quadratic
The equation is already simplified. It follows the general form of a quadratic equation:
\[
ax^2 + bx + c
\]
Where:
- \(a = -1\) (the coefficient of \(x^2\)),
- \(b = 2\) (the coefficient of \(x\)),
- \(c = 1\) (the constant).

### Step 2: Solving the quadratic equation

We can solve the equation by setting it equal to zero:

\[
-x^2 + 2x + 1 = 0
\]

This can be solved using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting \(a = -1\), \(b = 2\), and \(c = 1\):

\[
x = \frac{-2 \pm \sqrt{(2)^2 - 4(-1)(1)}}{2(-1)}
\]
\[
x = \frac{-2 \pm \sqrt{4 + 4}}{-2}
\]
\[
x = \frac{-2 \pm \sqrt{8}}{-2}
\]
\[
x = \frac{-2 \pm 2\sqrt{2}}{-2}
\]

### Step 3: Simplifying the solution
Now divide each term by \(-2\):

\[
x = 1 \pm \sqrt{2}
\]

Thus, the two possible solutions are:
\[
x_1 = 1 + \sqrt{2}, \quad x_2 = 1 - \sqrt{2}
\]

These are the roots of the quadratic equation.

Would you like more details or have any other questions?

Here are 5 related questions to expand on this concept:
1. What is the significance of the discriminant (\(b^2 - 4ac\)) in solving quadratics?
2. Can you factor the expression \(-x^2 + 2x + 1\) instead of using the quadratic formula?
3. How do we graph the quadratic function \(-x^2 + 2x + 1\)?
4. What are the applications of solving quadratic equations in real-life problems?
5. How can completing the square be used to solve this equation?

**Tip**: Always check if a quadratic equation can be factored easily before using the quadratic formula!

Learn how to solve the quadratic equation -x² + 2x + 1 step-by-step using the quadratic formula. Explore key algebraic concepts, including solving for roots and understanding the discriminant. Suitable for students in Grades 9-10.

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