To solve the inequality 2x - 6 + 5x(x - 3), we need to simplify the expression and then solve for x.

First, distribute the 5x to the terms inside the parentheses:
2x - 6 + 5x^2 - 15x

Combine like terms:
5x^2 - 13x - 6

Now, set the expression equal to zero:
5x^2 - 13x - 6 < 0

To solve this quadratic inequality, we can factor it:
(5x + 2)(x - 3) < 0

Now, we can use the zero product property to find the critical points:
5x + 2 = 0 --> x = -2/5
x - 3 = 0 --> x = 3

Plot these critical points on a number line:
-2/5 3

Now, we can test intervals to determine the sign of the expression:
For x < -2/5, choose x = -1:
(5(-1) + 2)(-1 - 3) = (-3)(-4) = 12 > 0

For -2/5 < x < 3, choose x = 0:
(5(0) + 2)(0 - 3) = (2)(-3) = -6 < 0

For x > 3, choose x = 4:
(5(4) + 2)(4 - 3) = (22)(1) = 22 > 0

From this, we can see that the expression is negative between -2/5 and 3, and positive outside of this interval.

Therefore, the solution to the inequality 2x - 6 + 5x(x - 3) < 0 is:
-2/5 < x < 3