Solve inequality x(-x-7)(-5x+2)<0

To solve the inequality x(-x-7)(-5x+2)<0, we first find the critical points by setting each factor equal to zero:

x = 0, -x - 7 = 0 \Rightarrow x = -7, and -5x + 2 = 0 \Rightarrow x = \frac{2}{5}.

Now, plot these critical points on a number line and test each interval:

1. Test the interval (-∞, -7): Pick any x < -7, say x = -8. Check x(-x-7)(-5x+2) which is negative, hence this interval satisfies the inequality.
2. Test the interval (-7, 0): Pick any -7 < x < 0, say x = -1. Check x(-x-7)(-5x+2) which is positive, hence this interval does not satisfy the inequality.
3. Test the interval (0, \frac{2}{5}): Pick any 0 < x < \frac{2}{5}, say x = \frac{1}{2}. Check x(-x-7)(-5x+2) which is negative, hence this interval satisfies the inequality.
4. Test the interval (\frac{2}{5}, ∞): Pick any \frac{2}{5} < x, say x = 1. Check x(-x-7)(-5x+2) which is positive, hence this interval does not satisfy the inequality.

Therefore, the solution to the inequality x(-x-7)(-5x+2)<0 is -7 < x < 0 or 0 < x < \frac{2}{5}.

\boxed{-7 < x < 0 \text{ or } 0 < x < \frac{2}{5}}