We are given the inequality:

\[
x^4 - 5x^3 < x^2 - 5x
\]

### Step 1: Bring all terms to one side
First, move all the terms to the left-hand side of the inequality to have it in the form \( f(x) < 0 \):

\[
x^4 - 5x^3 - x^2 + 5x < 0
\]

### Step 2: Factor the expression
Let's try factoring the expression on the left-hand side.

\[
x(x^3 - 5x^2 - x + 5) < 0
\]

Now we focus on factoring \( x^3 - 5x^2 - x + 5 \). We will use synthetic division or the rational root theorem to find roots. Testing \( x = 1 \) as a potential root:

\[
1^3 - 5(1)^2 - 1 + 5 = 1 - 5 - 1 + 5 = 0
\]

Since \( x = 1 \) is a root, we can divide \( x^3 - 5x^2 - x + 5 \) by \( x - 1 \). Using synthetic division, we get:

\[
x^3 - 5x^2 - x + 5 = (x - 1)(x^2 - 4x - 5)
\]

Next, factor \( x^2 - 4x - 5 \):

\[
x^2 - 4x - 5 = (x - 5)(x + 1)
\]

So, the complete factorization of the inequality becomes:

\[
x(x - 1)(x - 5)(x + 1) < 0
\]

### Step 3: Analyze the sign changes
We now solve \( x(x - 1)(x - 5)(x + 1) < 0 \).

The roots of the equation are \( x = 0, 1, -1, 5 \). These divide the real number line into intervals:

- \( (-\infty, -1) \)
- \( (-1, 0) \)
- \( (0, 1) \)
- \( (1, 5) \)
- \( (5, \infty) \)

We will test each interval to determine where the product is negative.

1. **For \( x \in (-\infty, -1) \)**: Pick \( x = -2 \), the expression is \( (-2)(-3)(-7)(-1) > 0 \).
2. **For \( x \in (-1, 0) \)**: Pick \( x = -\frac{1}{2} \), the expression is \( (-\frac{1}{2})(-\frac{3}{2})(-\frac{11}{2})(\frac{1}{2}) < 0 \).
3. **For \( x \in (0, 1) \)**: Pick \( x = \frac{1}{2} \), the expression is \( (\frac{1}{2})(-\frac{1}{2})(-\frac{9}{2})(\frac{3}{2}) > 0 \).
4. **For \( x \in (1, 5) \)**: Pick \( x = 3 \), the expression is \( (3)(2)(-2)(4) < 0 \).
5. **For \( x \in (5, \infty) \)**: Pick \( x = 6 \), the expression is \( (6)(5)(1)(7) > 0 \).

### Step 4: Write the solution
The solution to the inequality \( x(x - 1)(x - 5)(x + 1) < 0 \) is where the expression is negative, which occurs in the intervals:

\[
(-1, 0) \cup (1, 5)
\]

Thus, the solution to the inequality is:

\[
x \in (-1, 0) \cup (1, 5)
\]

Would you like more details on any specific step or have any questions?

### Related Questions:
1. How do you apply the rational root theorem to factor cubic polynomials?
2. What is synthetic division, and how does it work?
3. How do sign changes determine intervals in polynomial inequalities?
4. Can you always factor a cubic polynomial using simple factoring techniques?
5. How do we analyze inequalities when roots repeat?

### Tip:
For polynomial inequalities, analyzing the sign changes at critical points (roots) is key to determining the solution.

This guide provides a step-by-step solution to the inequality x^4 - 5x^3 < x^2 - 5x. Learn how to factor cubic polynomials using synthetic division and analyze sign changes to find the solution intervals. Suitable for students in Grades 10-12.

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