Given equation:
The given equation is x^2 + 2x + 1 = 0.

Factoring the equation:
We can factor the equation as follows:
(x + 1)(x + 1) = 0.

Solving for x:
To find the values of x, we set each factor equal to zero and solve for x:
x + 1 = 0.
x = -1.

Substituting x = -1:
Now, we substitute x = -1 back into the equation to check if it satisfies the equation:
(-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0.
Since the equation is satisfied, x = -1 is a valid solution.

Solving for p^3q^3:
To find the values of p^3q^3, we need to express x in terms of p and q.
Given that x = -1, we can write:
-1 = p + q.

Cubing both sides:
To find p^3q^3, we cube both sides of the equation:
(-1)^3 = (p + q)^3.
-1 = p^3 + 3p^2q + 3pq^2 + q^3.

Using the given equation:
We can substitute the values of p and q using the given equation x^2 + 2x + 1 = 0.
Since x = -1, we have:
(-1)^2 + 2(-1) + 1 = 0.
1 - 2 + 1 = 0.
0 = 0.

Substituting x = -1:
Now, we substitute x = -1 back into the equation -1 = p + q to solve for p and q:
-1 = p + q.
p = -1 - q.

Substituting p = -1 - q into the equation:
Substituting p = -1 - q into the equation p^3 + 3p^2q + 3pq^2 + q^3 = -1, we get:
(-1 - q)^3 + 3(-1 - q)^2q + 3(-1 - q)q^2 + q^3 = -1.

Simplifying the equation:
Expanding and simplifying the equation, we get:
-1 - 3q - 3q^2 - q^3 + 3q + 6q^2 + 3q^3 - 3q - 3q^2 + q^3 = -1.

Combining like terms:
Combining like terms, we get:
6q^2 + q^3 = 0.

Factoring the equation:
We can factor the equation as follows:
q^2(6 + q) = 0.

Solving for q:
Setting each factor equal to zero, we solve for q:
q^2 = 0 -> q = 0.
6 + q = 0 -> q = -6.

Finding p:
Substituting the values of q into the equation p = -1 -