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The equation provided is \( y = x^2 + 4x + 1 \), which is a quadratic equation. The vertex of a parabola given by \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:

\[
x = -\frac{b}{2a}
\]

In this case, the equation is \( y = x^2 + 4x + 1 \), where:
- \( a = 1 \)
- \( b = 4 \)

Now, let's find the x-coordinate of the vertex:

\[
x = -\frac{4}{2(1)} = -2
\]

To find the y-coordinate, we substitute \( x = -2 \) back into the equation:

\[
y = (-2)^2 + 4(-2) + 1 = 4 - 8 + 1 = -3
\]

So, the vertex is at \( (-2, -3) \).

Thus, the correct answer is \( \boxed{(-2, -3)} \).

Would you like more details or have any further questions?

Here are 5 related questions you might explore:
1. How do you find the axis of symmetry for a quadratic function?
2. What is the importance of the discriminant in a quadratic equation?
3. How does the value of \(a\) affect the direction of a parabola?
4. Can you convert the quadratic equation \( y = x^2 + 4x + 1 \) into vertex form?
5. How do you graph a quadratic function using its vertex and intercepts?

**Tip:** The vertex form of a quadratic equation is useful for identifying the vertex directly and is given by \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.

Which coordinate pair is the vertex of y = x^2 + 4x + 1?

Learn how to find the vertex of the quadratic function y = x^2 + 4x + 1. This step-by-step solution covers important algebraic concepts such as the vertex formula for parabolas. Ideal for students in Grades 9-11.

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