To solve the equation \( x(x-2) + 12 = (x+1)(x+2) \), we start by expanding both sides of the equation.

First, expand the left-hand side:
\[ x(x-2) + 12 = x^2 - 2x + 12 \]

Next, expand the right-hand side:
\[ (x+1)(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2 \]

Now, we have:
\[ x^2 - 2x + 12 = x^2 + 3x + 2 \]

Subtract \( x^2 \) from both sides to simplify:
\[ -2x + 12 = 3x + 2 \]

Next, isolate the \( x \) terms on one side by subtracting \( 3x \) from both sides:
\[ -2x - 3x + 12 = 2 \]
\[ -5x + 12 = 2 \]

Subtract 12 from both sides to further isolate the \( x \) term:
\[ -5x = 2 - 12 \]
\[ -5x = -10 \]

Finally, divide both sides by -5 to solve for \( x \):
\[ x = \frac{-10}{-5} \]
\[ x = 2 \]

So, the solution to the equation \( x(x-2) + 12 = (x+1)(x+2) \) is \( x = 2 \).