We are tasked with solving the inequality:

\[
3x^2 - 6x - 45 > 0
\]

### Step 1: Simplify the inequality
Factor out the greatest common factor (GCF), which is 3:

\[
3(x^2 - 2x - 15) > 0
\]

Now divide both sides of the inequality by 3:

\[
x^2 - 2x - 15 > 0
\]

### Step 2: Solve the corresponding quadratic equation
We solve the quadratic equation \(x^2 - 2x - 15 = 0\) to find its roots.

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)}
\]
\[
x = \frac{2 \pm \sqrt{4 + 60}}{2}
\]
\[
x = \frac{2 \pm \sqrt{64}}{2}
\]
\[
x = \frac{2 \pm 8}{2}
\]

Thus, the two solutions are:

\[
x = \frac{2 + 8}{2} = 5 \quad \text{and} \quad x = \frac{2 - 8}{2} = -3
\]

### Step 3: Analyze the sign of the quadratic
We now determine where the quadratic expression \(x^2 - 2x - 15 > 0\). The roots are \(x = -3\) and \(x = 5\), so the quadratic factors as:

\[
(x + 3)(x - 5) > 0
\]

This product is positive when both factors are either positive or both negative. We can check the sign of the expression in three intervals:
- \(x < -3\)
- \(-3 < x < 5\)
- \(x > 5\)

1. For \(x < -3\), both \((x + 3)\) and \((x - 5)\) are negative, so their product is positive.
2. For \(-3 < x < 5\), \((x + 3)\) is positive and \((x - 5)\) is negative, so the product is negative.
3. For \(x > 5\), both \((x + 3)\) and \((x - 5)\) are positive, so the product is positive.

### Step 4: Conclusion
The inequality \(x^2 - 2x - 15 > 0\) holds in the intervals:

\[
x \in (-\infty, -3) \cup (5, \infty)
\]

Thus, the solution to the inequality \(3x^2 - 6x - 45 > 0\) is:

\[
x \in (-\infty, -3) \cup (5, \infty)
\]

Would you like more details on any of the steps?

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Here are 5 related questions for further practice:
1. Solve the inequality \(2x^2 - 4x - 6 > 0\).
2. Solve the inequality \(x^2 - 9 > 0\).
3. Solve the quadratic equation \(x^2 - 3x + 2 = 0\).
4. Factor the quadratic expression \(x^2 + 6x + 9\).
5. Solve the inequality \(x^2 + 4x - 12 < 0\).

**Tip:** When solving inequalities involving quadratics, always check the sign of the factors in different intervals to determine where the inequality holds.

Learn how to solve the quadratic inequality 3x^2 - 6x - 45 > 0 with step-by-step instructions. This solution involves factoring, using the quadratic formula, and analyzing the sign of the expression over different intervals. Ideal for students in Grades 9-11 studying quadratic inequalities.

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