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Factoring polynomials is a key step when simplifying rational expressions. To factor polynomials, look for common factors in each term.
This makes complex expressions easier to work with and helps identify common terms.

In the given exercise, start by factoring the numerators and denominators. To factor a trinomial like \(v^2 + 9v + 20\), find two numbers that multiply to 20 and add up to 9. These numbers are 4 and 5, so we can write the numerator as \((v+4)(v+5)\).

Similarly, for the denominator \(v^2 + 10v + 24\), we need two numbers that multiply to 24 and add to 10, which are 4 and 6. Therefore, the denominator factors as \((v+4)(v+6)\).

Factoring helps break down complex expressions into their components, making it easier to manage and simplify.

In rational expressions, the numerator is the top part, while the denominator is the bottom part. Simplifying these parts individually can make handling the entire fraction easier.

The given exercise requires us to work with both numerators and denominators thoughtfully.

When dealings with a fraction like \(\frac{v^2+9v+20}{v^2+10v+24}\), the important point is to recognize and factor each part separately.

For the numerator \(v^2 + 9v + 20\), we factor it into \((v+4)(v+5)\), and for the denominator \(v^2 + 10v + 24\), it factors into \((v+4)(v+6)\).

By addressing numerators and denominators separately, you keep the process more organized, which leads to a clearer path to simplification.

Once you have factored both the numerators and denominators, you can cancel common terms. This will help simplify complex rational expressions.

In the exercise, for \(\frac{(v+4)(v+5)}{(v+4)(v+6)}\), both the numerator and denominator share a common factor, \(v+4\). Cancel these terms to simplify the fraction to \(\frac{v+5}{v+6}\).

Similarly, for the second fraction \(\frac{(z-8)(z+5)}{(z-8)(z+8)}\), the common factor \(z-8\) is shared between the numerator and denominator. Once you cancel them, you're left with \(\frac{z+5}{z+8}\).

Canceling common terms is a powerful step. It reduces the complexity of the expression and makes it easier to understand. Remember, you can only cancel factors (terms connected by multiplication), not terms connected by addition or subtraction.