Solving the Equation: (6x+2)6(x+2) = 2x(x7)(x+6) = 0
This equation presents a combination of linear and quadratic expressions. Let's break it down stepbystep to find the solutions.
Step 1: Simplify the Equation
First, we need to simplify both sides of the equation:

Left side: (6x+2)  6(x+2)
 Distribute the 6: 6x + 2  6x  12
 Combine like terms: 10

Right side: 2x(x7)(x+6)
 This is already simplified, but we can expand it for clarity.
Now, our equation looks like this: 10 = 2x(x7)(x+6)
Step 2: Solve for x
We need to find the values of x that satisfy the equation. Since the right side is a product of factors, we can use the Zero Product Property. This property states that if the product of several factors is zero, then at least one of the factors must be zero.
Therefore, we have two scenarios:

Scenario 1: 10 = 0 (This is impossible, so this scenario doesn't provide any solutions)

Scenario 2: 2x(x7)(x+6) = 0
 Using the Zero Product Property, we set each factor equal to zero:
 2x = 0 => x = 0
 x  7 = 0 => x = 7
 x + 6 = 0 => x = 6
 Using the Zero Product Property, we set each factor equal to zero:
Solutions
The solutions to the equation (6x+2)6(x+2) = 2x(x7)(x+6) = 0 are:
 x = 0
 x = 7
 x = 6