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Understanding how to divide polynomials is crucial for simplifying complex rational expressions. Although it might resemble long division with numbers, it involves a process of arranging terms in descending powers and then subtracting multiples of the divisor polynomial from the dividend polynomial.

For instance, when you're given a fraction of polynomials and asked to divide one by another, you rewrite this operation as a multiplication by taking the reciprocal of the divisor polynomial. This is demonstrated in our exercise by changing \(\frac{9 x^{2}+6 x+1}{x+5} \div \frac{3 x+1}{x^{2}+5 x}\) to \(\frac{9 x^{2}+6 x+1}{x+5} \times \frac{x^{2} + 5x}{3 x+1}\). This step eases the process and readies the expression for factoring and further simplification.

Factoring polynomials is akin to breaking down numbers into their prime factors - it is about expressing the polynomial as a product of its simplest polynomials. This method helps in simplifying rational expressions substantially because it can reveal common factors in the numerator and denominator that may be cancelled.

In the given solution, the numerator is a trinomial that looks like it could be perfect square trinomial because it fits the form \(a^2 + 2ab + b^2\), but after inspection, it's clear it cannot be simplified further. The denominator, however, contains a factor of \(x+5\), which is crucial for the step that follows.

The takeaway? Always look for the greatest common factor, the difference of squares, or trinomial patterns that might simplify to binomials to strip down the polynomials to their bones.

Dealing with rational expressions in algebra involves performing operations with fractions where the numerators and denominators are polynomials. The key is to remember that like numerical fractions, terms can be cancelled if they appear in both the numerator and the denominator — providing they are not just terms but factors.

In our example, after the change from division to multiplication and some factoring, we spot a \(x+5\) in both the numerator and denominator. Because \(x+5\) is a factor of the entire terms in which they appear, we're allowed to cancel them out, much like you would cancel out a 5 if you had \(\frac{5 \times 2}{5}\). This results in the simplified expression \(\frac{9x^2 + 6x + 1}{3x + 1}\) which is easier to handle.

Remember, the cancellation is valid across multiplication but never across addition or subtraction within a polynomial. Simplifying rational expressions therefore hinges on the ability to factor polynomials and apply the fundamental principles of fraction operations.