Now, we need to find the critical points by setting each factor equal to zero:
x^2 + 1 = 0 x^2 = -1 x = ±i
x^2 - 1 = 0 x^2 = 1 x = ±1
x + 5 = 0 x = -5
x - 3 = 0 x = 3
Now we can create intervals using these critical points (-∞, -5), (-5, -3), (-3, 1), (1, 3), (3, ∞) and test a point in each interval to determine the sign of the expression.
For example, testing x = -6 in the expression (x^4 + 1)(x^2 + 2x - 15) we get: ((-6)^4 + 1)((-6)^2 + 2(-6) - 15) = (1297)(9) > 0
Therefore, the solution to the inequality (x^4 + 1)(x^2 + 2x - 15) > 0 is: x ∈ (-∞, -5) U (-3, 1) U (3, ∞)
To solve this inequality, we first need to factor the expression on the left side:
(x^4 + 1)(x^2 + 2x - 15) = (x^2 + 1)(x^2 - 1)(x + 5)(x - 3)
Now, we need to find the critical points by setting each factor equal to zero:
x^2 + 1 = 0
x^2 = -1
x = ±i
x^2 - 1 = 0
x^2 = 1
x = ±1
x + 5 = 0
x = -5
x - 3 = 0
x = 3
Now we can create intervals using these critical points (-∞, -5), (-5, -3), (-3, 1), (1, 3), (3, ∞) and test a point in each interval to determine the sign of the expression.
For example, testing x = -6 in the expression (x^4 + 1)(x^2 + 2x - 15) we get:
((-6)^4 + 1)((-6)^2 + 2(-6) - 15) = (1297)(9) > 0
Therefore, the solution to the inequality (x^4 + 1)(x^2 + 2x - 15) > 0 is:
x ∈ (-∞, -5) U (-3, 1) U (3, ∞)