If x^4 + x^3 + x^2 + x + 1 = 0 what is the value of the product (x1 + 1)(x2 + 1)(x3 +1)(x4 + 1) ?

The roots of the given equation are:x1, x2, x3,
x4.

We'll use Viete's
relations:

x1 + x2 + x3 + x4 = -b/a =
-1

x1*x2 + x1*x3 + x1*x4 + x2*x3 + x2*x4 + x3*x4 = c/a =
1

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 = -d/a =
-1

x1*x2*x3*x4 = e/a = 1

We'll
calculate the product:

P =
(1+x1)(1+x2)(1+x3)(1+x4)

P = 1+(x1 + x2 + x3 + x4) + (x1*x2
+ x1*x3 + x1*x4 + x2*x3 + x2*x4 + x3*x4) + (x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4) +
(x1*x2*x3*x4)

P = 1 - b/a + c/a - d/a + e/a = 1-1+1-1+1 =
1

P =
1