To solve the equation, we can start by isolating the absolute value term on one side of the equation.

|3x + 6| - 12 = x + 1

Next, we can split the equation into two cases, one for when the expression inside the absolute value is positive and one for when it is negative.

Case 1: 3x + 6 is positive
In this case, the absolute value can be removed, so we have:

3x + 6 - 12 = x + 1

Simplifying the equation:

3x - 6 = x + 1

Subtracting x from both sides:

2x - 6 = 1

Adding 6 to both sides:

2x = 7

Dividing by 2:

x = 7/2

Case 2: 3x + 6 is negative
In this case, the expression inside the absolute value becomes its negation, so we have:

-(3x + 6) - 12 = x + 1

Simplifying the equation:

-3x - 6 - 12 = x + 1

Combining like terms:

-3x - 18 = x + 1

Adding 3x to both sides:

-18 = 4x + 1

Subtracting 1 from both sides:

-19 = 4x

Dividing by 4:

x = -19/4

Therefore, the solutions to the equation are x = 7/2 and x = -19/4.