Let's solve the equation step by step:

Given equation:
\[
5x(x + 6) - 2x - 12 = 0
\]

### Step 1: Distribute the \(5x\) through the terms in the parentheses.
\[
5x^2 + 30x - 2x - 12 = 0
\]

### Step 2: Combine like terms.
\[
5x^2 + 28x - 12 = 0
\]

### Step 3: Solve the quadratic equation using the quadratic formula.
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 5\), \(b = 28\), and \(c = -12\).

### Step 4: Plug the values into the quadratic formula.
\[
x = \frac{-28 \pm \sqrt{28^2 - 4(5)(-12)}}{2(5)}
\]
\[
x = \frac{-28 \pm \sqrt{784 + 240}}{10}
\]
\[
x = \frac{-28 \pm \sqrt{1024}}{10}
\]
\[
x = \frac{-28 \pm 32}{10}
\]

### Step 5: Simplify the solutions.
1. \(x_1 = \frac{-28 + 32}{10} = \frac{4}{10} = 0.4\)
2. \(x_2 = \frac{-28 - 32}{10} = \frac{-60}{10} = -6\)

So, the solutions are:
\[
x = 0.4 \quad \text{or} \quad x = -6
\]

Would you like more details or have any questions?

Here are five related questions you might consider:
1. How would you solve the quadratic equation \(x^2 + 7x + 10 = 0\)?
2. What is the discriminant of a quadratic equation, and how does it determine the nature of the roots?
3. How would you factor the quadratic equation \(x^2 - 5x + 6 = 0\)?
4. What are the steps to complete the square for the quadratic equation \(3x^2 + 12x + 5 = 0\)?
5. How do you graph the quadratic function \(y = 2x^2 + 4x - 6\)?

**Tip:** When solving quadratic equations, always check if the quadratic can be factored easily before using the quadratic formula. This can save time and avoid calculation errors.

Learn how to solve the quadratic equation 5x(x + 6) - 2x - 12 = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and factoring. Suitable for students in Grades 9-12.

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