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

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To solve this equation, we can break it down into two separate quadratic equations:

Let's denote:
a = x^2 + 3x
b = 3

Then the first quadratic equation becomes:
a + 3 = 0
a = -3

The second quadratic equation becomes:
a + 1 = 0
a = -1

Therefore, the solutions for x are the roots of the quadratic equations -3 and -1:
x = (-3 - 3) / 2 = -6 / 2 = -3
x = (-3 - 1) / 2 = -4 / 2 = -2

So the solutions for the given equation are x = -3 and x = -2.