The inequality given in the image is:

$-x^2 + x + 6 > 0$

Let's solve this step-by-step:

Step 1: Rearrange the inequality

The inequality is already in standard form. To make it easier to solve, we can multiply both sides of the inequality by $-1$, which will reverse the inequality sign:

$x^2 - x - 6 < 0$

Now, we will solve the inequality $x^2 - x - 6 < 0$.

Step 2: Solve the related quadratic equation

First, solve the quadratic equation $x^2 - x - 6 = 0$. This can be factored:

$x^2 - x - 6 = (x - 3)(x + 2) = 0$

The solutions to this equation are:

$x = 3 \quad \text{and} \quad x = -2$

Step 3: Analyze the sign of the quadratic expression

Now, we analyze the sign of $(x - 3)(x + 2)$ for different intervals of $x$, based on the roots $-2$ and $3$. The intervals are:

• $x < -2$
• $-2 < x < 3$
• $x > 3$

We test the sign of $(x - 3)(x + 2)$ in each interval:

• For $x < -2$ (e.g., $x = -3$): $(x - 3) < 0$ and $(x + 2) < 0$, so $(x - 3)(x + 2) > 0$
• For $-2 < x < 3$ (e.g., $x = 0$): $(x - 3) < 0$ and $(x + 2) > 0$, so $(x - 3)(x + 2) < 0$
• For $x > 3$ (e.g., $x = 4$): $(x - 3) > 0$ and $(x + 2) > 0$, so $(x - 3)(x + 2) > 0$

Step 4: Solution to the inequality

We want the quadratic expression to be negative, so we select the interval where $(x - 3)(x + 2) < 0$, which is:

$-2 < x < 3$

Final Answer:

The solution to the inequality $-x^2 + x + 6 > 0$ is:

$x \in (-2, 3)$

Would you like more details or explanations on any step?

Here are 5 related questions you might want to explore:

1. How do we solve quadratic inequalities in general?
2. What is the geometric interpretation of this inequality on a graph?
3. Can we apply the quadratic formula instead of factoring?
4. What happens if the quadratic inequality had $\geq$ instead of $>$?
5. How do we check the solution using a test point method?

Tip: Always remember that when multiplying or dividing an inequality by a negative number, the direction of the inequality sign reverses!