To solve the equation cot(2x + π/4) = -1, we need to find the values of x that satisfy this equation.

First, let's recall the definition of cotangent:

cot(θ) = 1/tan(θ)

So, we can rewrite the equation as:

1/tan(2x + π/4) = -1

Next, let's recall the definition of tangent:

tan(θ) = sin(θ)/cos(θ)

Using this definition, we can rewrite the equation as:

1/(sin(2x + π/4)/cos(2x + π/4)) = -1

Now, let's simplify the equation:

cos(2x + π/4) = -sin(2x + π/4)

Using the trigonometric identity cos(θ) = -sin(θ + π/2), we can rewrite the equation as:

cos(2x + π/4) = sin(2x + π/4 + π/2)

Now, let's simplify further:

cos(2x + π/4) = sin(2x + 3π/4)

Since the cosine and sine functions are equal at π/4, we can set the arguments of the functions equal to each other:

2x + π/4 = 2x + 3π/4

Subtracting 2x from both sides, we get:

π/4 = 3π/4

This equation is not possible since π/4 is not equal to 3π/4.

Therefore, there are no solutions to the equation cot(2x + π/4) = -1.