Now, we need to find the roots of the equation by setting it equal to zero:
x^4 - 6x^3 - 5x^2 - 24x - 117 = 0
This inequality is a bit difficult to solve without the use of a computer or calculator due to the quartic equation involved. However, the roots can be found using methods like numerical analysis or computer software.
To solve this inequality, we need to first square both sides of the inequality to get rid of the square root:
(x-3)^2(x^2 + 4) ≤ (x^2 + 9)^2
Expanding both sides gives:
(x^2 - 6x + 9)(x^2 + 4) ≤ x^4 + 18x^2 + 81
Now, simplify:
x^4 + 4x^2 - 6x^3 - 24x + 9x^2 - 36 + 4x^2 ≤ x^4 + 18x^2 + 81
Combine like terms:
4x^2 - 6x^3 - 24x + 9x^2 - 36 + 4x^2 ≤ x^4 + 18x^2 + 81
13x^2 - 6x^3 - 24x - 36 ≤ x^4 + 18x^2 + 81
Rearranging to standard form:
x^4 - 6x^3 - 5x^2 - 24x - 117 ≥ 0
Now, we need to find the roots of the equation by setting it equal to zero:
x^4 - 6x^3 - 5x^2 - 24x - 117 = 0
This inequality is a bit difficult to solve without the use of a computer or calculator due to the quartic equation involved. However, the roots can be found using methods like numerical analysis or computer software.