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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 points to determine when the expression is greater than zero.

First, let's find the critical points by setting the expression equal to zero: x(6+x)(x-4/3) = 0

This expression will equal zero when: x = 0, x = -6, x = 4/3

Now, we will test the intervals created by these critical points to determine when the expression is greater than zero.

Interval 1: x < -6 Choose x = -7: (-7)(-1)(-7-4/3) = (-7)(-1)(-25/3) = 175/3 > 0

Interval 2: -6 < x < 0 Choose x = -1: (-1)(5)(-1-4/3) = (-1)(5)(-7/3) = 35/3 > 0

Interval 3: 0 < x < 4/3 Choose x = 1: (1)(7)(1-4/3) = (1)(7)(-1/3) = -7/3 < 0

Interval 4: x > 4/3 Choose x = 2: (2)(8)(2-4/3) = (2)(8)(2/3) = 32/3 > 0

Therefore, the solution to the inequality x(6+x)(x-4/3) > 0 is x < -6 or -6 < x < 0 or x > 4/3.