To write the given equation in standard form, let's start by simplifying and arranging it.

The equation provided is: $-3x^2 + 2x + 6 = -x^2$

1. Move $-x^2$ from the right side to the left side of the equation: $-3x^2 + 2x + 6 + x^2 = 0$

2. Combine like terms: $(-3x^2 + x^2) + 2x + 6 = 0$ This simplifies to: $-2x^2 + 2x + 6 = 0$

3. To put it in standard form $ax^2 + bx + c = 0$, rewrite it as: $-2x^2 + 2x + 6 = 0$

The quadratic equation in standard form is: $-2x^2 + 2x + 6 = 0$

Would you like further details on solving this equation or graphing it?

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1. What is the vertex form of this quadratic equation?
2. How can we determine if this quadratic opens upwards or downwards?
3. What are the solutions to this quadratic equation?
4. How does changing the sign of the coefficient of $x^2$ affect the graph?
5. What is the axis of symmetry for this quadratic equation?

Tip: For any quadratic equation in the form $ax^2 + bx + c = 0$, the sign of $a$ determines the direction of the parabola's opening.