The inequality given is:

$(-x + 1) \cdot (2x + 3)^2 \cdot (4x - 5)^4 > 0$

To solve this inequality, let's analyze each factor:

1. Factor 1: $-x + 1$

   • This factor changes sign at $x = 1$.
   • For $x < 1$, $-x + 1 > 0$.
   • For $x > 1$, $-x + 1 < 0$.

2. Factor 2: $(2x + 3)^2$

   • Since this is a square, $(2x + 3)^2 \geq 0$ for all $x$, and it equals 0 when $2x + 3 = 0$, i.e., $x = -\frac{3}{2}$.
   • For any other value of $x$, $(2x + 3)^2 > 0$.

3. Factor 3: $(4x - 5)^4$

   • Since this is a fourth power, $(4x - 5)^4 \geq 0$ for all $x$, and it equals 0 when $4x - 5 = 0$, i.e., $x = \frac{5}{4}$.
   • For any other value of $x$, $(4x - 5)^4 > 0$.

Now, consider the product of these factors:

Sign Analysis:

• At $x = 1$: The first factor $-x + 1 = 0$, and thus the product is 0, not positive.
• At $x = -\frac{3}{2}$: The second factor $(2x + 3)^2 = 0$, and thus the product is 0, not positive.
• At $x = \frac{5}{4}$: The third factor $(4x - 5)^4 = 0$, and thus the product is 0, not positive.

For the product to be positive:

1. The first factor must be positive, meaning $x < 1$.
2. The second factor is positive for all $x \neq -\frac{3}{2}$.
3. The third factor is positive for all $x \neq \frac{5}{4}$.

Solution Set:

To satisfy all conditions where the product is positive:

• $x < 1$ (since $-x + 1 > 0$).
• Exclude $x = -\frac{3}{2}$ and $x = \frac{5}{4}$ since these make the product zero.

Thus, the solution to the inequality is:

$x \in \left(-\infty, -\frac{3}{2}\right) \cup \left(-\frac{3}{2}, \frac{5}{4}\right)$

This is the interval where the expression is greater than zero.

Would you like more details on any specific part? Here are some related questions you might find helpful:

1. What is the general approach for solving polynomial inequalities?
2. How does the sign of each factor affect the overall product in inequalities?
3. Why does squaring or taking the fourth power of an expression always yield a non-negative result?
4. How would the solution change if the inequality were $\geq 0$ instead of $> 0$?
5. Can you explain how to graph the solution set on a number line?

Tip: When solving inequalities, always consider the sign of each factor and where each factor equals zero. This will help you determine the intervals where the inequality holds.