### Instruction: Solve the inequality
\[-12x^2 + 3x - 5 < 0.\].### Response: To solve this quadratic inequality, I want to find the values of x that make the expression negative.
I notice that the leading coefficient of the quadratic is negative, so the graph of the function is a downward-opening parabola.
That means the expression is negative between the two x-intercepts of the parabola, if they exist.
To find the x-intercepts, I can set the expression equal to zero and use the quadratic formula:
\[x = \frac{-3 \pm \sqrt{3^2 - 4(-12)(-5)}}{2(-12)}.\]
Simplifying, I get
\[x = \frac{-3 \pm \sqrt{9 - 240}}{-24}.\]
Since the radicand is negative, there are no real solutions for x. This means the parabola never crosses the x-axis, and the expression is always negative for any real value of x.
So the solution to the inequality is the entire real number line, or \(-\infty < x < \infty\).