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To solve this inequality, we need to find the critical points where the expression equals zero and then test the intervals between these critical points to determine where the expression is greater than or equal to zero.

First, let's find the critical points by setting the expression equal to zero and solving for x:

-x^6 + 3x^5 - 37x^4 + 8x^3 + 22x^2 + 72x - 88 = 0

We can factor out an x^4 from the expression to simplify it:

x^4(-x^2 + 3x - 37) + 8x^3 + 22x^2 + 72x - 88 = 0

Now, we can solve for x by setting each factor equal to zero:

x^4 = 0 x = 0

-x^2 + 3x - 37 = 0 Using the quadratic formula, we get: x = (-(3) ± sqrt((3)^2 - 4(-1)(-37))) / 2(-1) x = (-(3) ± sqrt(9 - 148)) / -2 x = (-(3) ± sqrt(-139)) / -2 Since the square root of a negative number is not real, there are no real solutions for this part of the equation.

Therefore, the only critical point is x = 0.

Next, we need to test the intervals created by the critical point x = 0 to determine where the expression is greater than or equal to zero.

We can choose test points x = -1, x = 1, and x = 2 to test the intervals (-∞, 0), (0, ∞).

For x = -1: -x^6 + 3x^5 - 37x^4 + 8x^3 + 22x^2 + 72x - 88 = -1 + 3 + 37 + 8 + 22 - 72 - 88 = -91 Since -91 < 0, the expression is negative for x = -1.

For x = 1: -x^6 + 3x^5 - 37x^4 + 8x^3 + 22x^2 + 72x - 88 = -1 + 3 - 37 + 8 + 22 + 72 - 88 = -21 Since -21 < 0, the expression is negative for x = 1.

For x = 2: -x^6 + 3x^5 - 37x^4 + 8x^3 + 22x^2 + 72x - 88 = -64 + 96 - 592 + 64 + 88 + 144 - 88 = 28 Since 28 > 0, the expression is positive for x = 2.

Therefore, the solution to the inequality -x^6 + 3x^5 - 37x^4 + 8x^3 + 22x^2 + 72x - 88 >= 0 is x ∈ [2, ∞).