-180° ≤ x ≤ 0°

The above implies that
-720° ≤ 4x ≤ 0° ==> this is needed for the angle 4x later

-540° ≤ 3x ≤ 0°
-570° ≤ 3x - 30° ≤ -30° ==> this is needed for the angle 3x - 30° later

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cot 4x sin (3x - 30°) = 0

This means that
cot 4x = 0 or sin (3x - 30°) = 0

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For cot 4x = 0,

1 / tan 4x = 0
tan 4x = infinity

This occurs at the points where the tangent graphs are undefined (90, 270, 450, 630, ... and -90, -270, -450, -630, ...)

We need our 4x to lie between -720° and 0° as explained above, so

4x = -630°, -450°, -270°, -90°
x = -157.5°, -112.5°, -67.5°, -22.5°

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For sin (3x - 30°) = 0 (bear in mind that we need our angle 3x - 30° to lie between and including -570° and -30°)...

3x - 30° = -540°, -360°, -180° (these start from 0 in the base case and always increase or decrease in increments of 180°)

3x = -510°, -330°, -150°
x = -170°, -110°, -50°

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Putting them together,

x = -170°, -157.5°, -112.5°, -110°, -67.5°, -50°, -22.5°