To solve the equation sin(2x + π/4) + cosx = 0, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1.

First, let's rewrite the equation using this identity:

sin(2x + π/4) + cosx = 0
sin(2x + π/4) = -cosx

Now, square both sides of the equation:

sin^2(2x + π/4) = cos^2(x)

Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:

1 - cos^2(2x + π/4) = cos^2(x)

Now, let's simplify the equation:

1 - cos^2(2x + π/4) - cos^2(x) = 0

Using the double angle formula for cosine, cos(2x) = 2cos^2(x) - 1, we can rewrite the equation as:

1 - (2cos^2(x) - 1) - cos^2(x) = 0
1 - 2cos^2(x) + 1 - cos^2(x) = 0
-3cos^2(x) + 2 = 0

Now, solve for cos^2(x):

3cos^2(x) = 2
cos^2(x) = 2/3

Taking the square root of both sides, we get:

cos(x) = ±√(2/3)

Now, we can find the values of x that satisfy the equation by taking the inverse cosine of both sides:

x = ±cos^(-1)(√(2/3))

Therefore, the solutions to the equation sin(2x + π/4) + cosx = 0 are:

x = ±cos^(-1)(√(2/3)) + 2πn, where n is an integer.