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When simplifying rational expressions, one of the first steps is factoring any polynomials in the numerator and denominator. Factoring involves rewriting a polynomial as a product of its simpler factors. For instance, consider the polynomial in the numerator from the problem above: \( q^2 - 4q \). To factor this, look for common factors in each term of the polynomial. Notice that both terms share a common factor of \( q \). Thus, \( q^2 - 4q \) can be written as \( q(q - 4) \). Similarly, for the denominator \( -q^2 + 4q \), we can factor out \( -q \): \( -q(q - 4) \). Once the polynomials are factored, it becomes easier to identify common factors in the numerator and denominator.

Common factors play a vital role in simplifying rational expressions. In the previous step, we factored the numerator and denominator to find their common factors. Now, let's look at the simplified rational expression: \( \frac{q(q - 4)}{-q(q - 4)} \). Here, \( q - 4 \) is a common factor in both the numerator and denominator. Since we are dividing the same term by itself, it cancels out: \( \frac{q(q - 4)}{-q(q - 4)} = \frac{q}{-q} \). Identifying and canceling out these common factors simplifies the expression significantly.

The goal of simplifying rational expressions is to write them in their lowest terms. This means removing all common factors from the numerator and denominator. Once common factors like \( q - 4 \) are canceled out in the expression \( \frac{q(q - 4)}{-q(q - 4)} \), we are left with \( \frac{q}{-q} \). In this fraction, \( q \) is a common factor that can also be removed, leaving us with \( -1 \): \( \frac{q}{-q} = -1 \). Thus, \( -1 \) is the simplest form, or lowest terms, of the original rational expression. Simplifying to lowest terms helps in making expressions more manageable and easier to handle in further calculations.