To solve the inequality −3x^2 + 4x − 1 < 0, we first need to find the critical points by setting the expression equal to zero:

−3x^2 + 4x − 1 = 0

This is a quadratic equation that can be solved using the quadratic formula:

x = [ -b ± √(b^2 - 4ac) ] / 2a

In this case, a = -3, b = 4, and c = -1. Plugging these values into the formula, we get:

x = [ -4 ± √(4^2 - 4(-3)(-1)) ] / 2(-3)
x = [ -4 ± √(16 - 12) ] / -6
x = [ -4 ± √4 ] / -6
x = [ -4 ± 2 ] / -6

This gives us two possible critical points:

x = (-4 + 2) / -6 = -2 / -6 = 1/3
x = (-4 - 2) / -6 = -6 / -6 = 1

Now we can test the intervals created by these critical points to determine where the inequality is true. We choose test points within each interval and plug them into the original inequality:

For x < 1/3, we can choose x = 0:
−3(0)^2 + 4(0) − 1 < 0
-1 < 0 (true)

For 1/3 < x < 1, we can choose x = 1/2:
−3(1/2)^2 + 4(1/2) − 1 < 0
-3/4 + 2 - 1 < 0
-3/4 + 1 < 0
1/4 < 0 (false)

For x > 1, we can choose x = 2:
−3(2)^2 + 4(2) − 1 < 0
-12 + 8 - 1 < 0
-5 < 0 (true)

Therefore, the solution to the inequality −3x^2 + 4x − 1 < 0 is x < 1/3 or x > 1.