To solve the equation, we can start by expanding the square terms:

(x - 4)^2 - (2x + 1)^2 = 0

(x^2 - 8x + 16) - (4x^2 + 4x + 1) = 0

Now, let's simplify the equation by combining like terms:

x^2 - 8x + 16 - 4x^2 - 4x - 1 = 0

-3x^2 - 12x + 15 = 0

Next, let's move all the terms to one side of the equation:

-3x^2 - 12x + 15 = 0

3x^2 + 12x - 15 = 0

Now, let's factor the quadratic equation:

3(x^2 + 4x - 5) = 0

Next, let's solve for x by setting each factor equal to zero:

x^2 + 4x - 5 = 0

(x + 5)(x - 1) = 0

Setting each factor equal to zero, we have:

x + 5 = 0 or x - 1 = 0

Solving for x in each equation, we get:

x = -5 or x = 1

Therefore, the solutions to the equation are x = -5 and x = 1.