To solve this equation, we first need to expand the left side expression:

(x+4)^2 = (x+4)(x+4) = x^2 + 8x + 16

(x+1)(x-2) = x^2 - 2x + x - 2 = x^2 - x - 2

Now, we can substitute these expanded expressions back into the original equation and simplify:

x^2 + 8x + 16 - (x^2 - x - 2) = 2x - 3

x^2 + 8x + 16 - x^2 + x + 2 = 2x - 3

8x + 16 + x + 2 = 2x - 3

9x + 18 = 2x - 3

Subtract 2x from both sides:

7x + 18 = -3

Subtract 18 from both sides:

7x = -21

Divide by 7:

x = -3

Therefore, the solution to the equation is x = -3.