To solve the equation \((x - 3)(x + 2) + 9x = 3(x^2 - 5) - 1\), we will first expand both sides and then simplify the equation.

### Step 1: Expand both sides
- Expand \((x - 3)(x + 2)\):
  \[
  (x - 3)(x + 2) = x^2 + 2x - 3x - 6 = x^2 - x - 6
  \]
- The equation now becomes:
  \[
  x^2 - x - 6 + 9x = 3(x^2 - 5) - 1
  \]

### Step 2: Simplify both sides
- Combine like terms on the left side:
  \[
  x^2 - x + 9x - 6 = x^2 + 8x - 6
  \]
- Expand and simplify the right side:
  \[
  3(x^2 - 5) - 1 = 3x^2 - 15 - 1 = 3x^2 - 16
  \]
- Now the equation looks like:
  \[
  x^2 + 8x - 6 = 3x^2 - 16
  \]

### Step 3: Move all terms to one side of the equation
- Subtract \(3x^2\) from both sides:
  \[
  x^2 + 8x - 6 - 3x^2 + 16 = 0
  \]
- Combine like terms:
  \[
  -2x^2 + 8x + 10 = 0
  \]
  
### Step 4: Solve the quadratic equation
The quadratic equation is \(-2x^2 + 8x + 10 = 0\). We can solve this using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = -2\), \(b = 8\), and \(c = 10\).

### Step 5: Calculate the discriminant
\[
\text{Discriminant} = b^2 - 4ac = 8^2 - 4(-2)(10) = 64 + 80 = 144
\]

### Step 6: Find the roots
\[
x = \frac{-8 \pm \sqrt{144}}{2(-2)} = \frac{-8 \pm 12}{-4}
\]

### Step 7: Solve for \(x\)
1. \(x_1 = \frac{-8 + 12}{-4} = \frac{4}{-4} = -1\)
2. \(x_2 = \frac{-8 - 12}{-4} = \frac{-20}{-4} = 5\)

### Final Answer:
The solutions to the equation are \(x = -1\) and \(x = 5\).

---

Would you like any further details or have any questions?

### Related Questions:
1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in a quadratic equation?
3. Can you graph the quadratic equation \(-2x^2 + 8x + 10\) and interpret the roots?
4. What is the impact of coefficients on the shape of a quadratic graph?
5. How can you solve quadratic equations using factoring instead of the quadratic formula?

### Tip:
Always double-check your work by substituting the solutions back into the original equation to verify correctness!

Learn how to solve the equation (x - 3)(x + 2) + 9x = 3(x^2 - 5) - 1 with a detailed step-by-step solution. This problem involves algebraic equations, quadratic formulas, and equation simplification. Suitable for advanced high school students (Grades 10-12).

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