The expression provided in the image is a fraction with two rational expressions. The entire expression reads:

$\frac{6x^4 - 30x^3}{6x^2 - 54x + 120} \times \frac{x^2 - 5x + 4}{x^3 - 8x^2 + 7x}$

Let's break it down step by step:

1. Simplify each part of the expression.

• First rational expression: $\frac{6x^4 - 30x^3}{6x^2 - 54x + 120}$ Factor both the numerator and denominator:

  • The numerator $6x^4 - 30x^3 = 6x^3(x - 5)$.
  • The denominator $6x^2 - 54x + 120 = 6(x^2 - 9x + 20)$. Now, we factor $x^2 - 9x + 20$: $x^2 - 9x + 20 = (x - 5)(x - 4)$ So the first expression becomes: $\frac{6x^3(x - 5)}{6(x - 5)(x - 4)} = \frac{x^3}{x - 4} \quad (\text{cancel } 6(x - 5))$

• Second rational expression: $\frac{x^2 - 5x + 4}{x^3 - 8x^2 + 7x}$ Factor both the numerator and the denominator:

  • The numerator $x^2 - 5x + 4$ factors as $(x - 1)(x - 4)$.
  • The denominator $x^3 - 8x^2 + 7x = x(x^2 - 8x + 7)$, and $x^2 - 8x + 7$ factors as $(x - 1)(x - 7)$. So the second expression becomes: $\frac{(x - 1)(x - 4)}{x(x - 1)(x - 7)}$ Cancel out the common factor $(x - 1)$, leaving: $\frac{x - 4}{x(x - 7)}$

2. Combine the two simplified expressions:

Now, multiply the two simplified expressions: $\frac{x^3}{x - 4} \times \frac{x - 4}{x(x - 7)}$ Cancel the common $(x - 4)$ term: $\frac{x^3}{x(x - 7)} = \frac{x^2}{x - 7}$

Thus, the final simplified expression is: $\frac{x^2}{x - 7}$

Would you like further details or have any questions on this process?

Here are 5 related questions you could explore:

1. How would the solution change if there were no common factors to cancel?
2. Can you factor a cubic polynomial if it doesn't have any real roots?
3. What techniques are used to simplify complex rational expressions?
4. How do common denominators work in rational expressions with addition or subtraction?
5. What is the process to check if a factorization is correct?

Tip: Always look for common factors first when simplifying rational expressions. It can significantly reduce the complexity of the problem.