(x + 17) / (x ² + 3) ≥ 4
x + 17 ≥ 4 (x ² + 3)
4x ² - x - 5 ≤ 0
(4x - 5) (x + 1) ≤ 0
The left side is equal to 0 when either
4x - 5 = 0 or x + 1 = 0
x = 5/4 or x = -1
For x < -1, we have both x + 1 < 0 and 4x - 5 < 0, so their product is positive and there are no solutions to the inequality in this interval.
For -1 < x < 5/4, we have x + 1 > 0 and 4x - 5 < 0, so their product is negative.
For x > 5/4, we have both x + 1 > 0 and 4x - 5 > 0, giving a positive product and thus no additional solutions.
The solution set is then the second interval, but including the endpoints since that makes the expression on the left equal to 0:
-1 ≤ x ≤ 5/4