To solve the inequality 2x(x²-x-3) < 0, we can start by factoring the quadratic expression inside the parentheses.

The quadratic expression x²-x-3 can be factored as (x-3)(x+1).

Now we have the inequality 2x(x-3)(x+1) < 0.

To find the values of x that satisfy this inequality, we can use the concept of intervals.

First, we can find the critical points by setting each factor equal to zero:

x = 0, x = 3, x = -1

These critical points divide the number line into four intervals: (-∞, -1), (-1, 0), (0, 3), and (3, ∞).

Now we can test a value from each interval to determine if it satisfies the inequality.

For the interval (-∞, -1), we can choose x = -2:

2(-2)(-2-3)(-2+1) = 2(-2)(-5)(-1) = 20 > 0

Since the expression is positive, this interval does not satisfy the inequality.

For the interval (-1, 0), we can choose x = -0.5:

2(-0.5)(-0.5-3)(-0.5+1) = 2(-0.5)(-3.5)(0.5) = 1.75 > 0

Again, the expression is positive, so this interval does not satisfy the inequality.

For the interval (0, 3), we can choose x = 1:

2(1)(1-3)(1+1) = 2(1)(-2)(2) = -8 < 0

Finally, we have found an interval where the expression is negative. This interval satisfies the inequality.

For the interval (3, ∞), we can choose x = 4:

2(4)(4-3)(4+1) = 2(4)(1)(5) = 40 > 0

Once again, the expression is positive, so this interval does not satisfy the inequality.

Therefore, the solution to the inequality 2x(x²-x-3) < 0 is x ∈ (0, 3).