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Understanding how to multiply fractions is a fundamental math skill. To multiply two fractions, simply follow these steps. First, multiply the numerators (top numbers) of the fractions together to get the new numerator. Then, multiply the denominators (bottom numbers) of the fractions to obtain the new denominator. The product of two fractions is another fraction formed by this new numerator over the new denominator.

For example, if we take the exercise \( -\frac{5 h k^{3}}{9} \cdot \frac{3}{4 h} \), we would start by multiplying the numerators \( -5hk^3 \) and \( 3 \) to get \( -15hk^3 \) for our new numerator. Similarly, we multiply the denominators \( 9 \) and \( 4h \) to get \( 36h \) for our new denominator, resulting in the fraction \( -\frac{15hk^3}{36h} \). Just like that, you've multiplied two fractions!

Once you have multiplied the fractions and combined the numerator and denominator, the next step is to express the fraction in the lowest terms. This process simplifies the fraction to its simplest form where the numerator and denominator have no common factors other than 1. To do this, find the greatest common divisor of both the numerator and denominator and divide them both by this number.

In the given exercise, the fraction produced is \( -\frac{15hk^3}{36h} \). To simplify, we look for common factors in the numerator and denominator. Both contain a common factor of \( h \) and the number \( 3 \) is a common divisor of \( 15 \) and \( 36 \). Dividing the numerator and denominator by these common factors results in the fraction \( -\frac{5k^3}{12} \)—our simplified fraction in the lowest terms.

Common factors play a crucial role in simplifying fractions. A 'common factor' is simply a number that divides evenly into two or more other numbers. When simplifying a fraction to its lowest terms, identifying and dividing the numerator and denominator by their common factors reduces the fraction to its simplest form.

Here’s how you identify common factors: First, list out the factors of both the numerator and the denominator, and look for numbers that appear in both lists. In our exercise, the common factors for the numerator \( -15hk^3 \) and the denominator \( 36h \) are \( 3 \) and \( h \), as both \( 15 \) and \( 36 \) are divisible by \( 3 \) and they both have \( h \) as a factor. Once you've found the common factors, you can divide them out to simplify the fraction. It's an essential skill for mathematical proficiency and is used not just in fractions, but also for simplifying algebraic expressions and solving equations.