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When working with fractions, having a common denominator is crucial for simplifying expressions. A common denominator is a shared multiple of the denominators of two or more fractions. In our exercise, both fractions have the same denominator: \(x-6\).
This means we can directly perform the operation on the numerators without any additional steps.
Here is why a common denominator is helpful:

• It allows us to compare or combine fractions without changing their values.
• It simplifies calculations, as there's no need to find equivalent fractions.

If the denominators were not the same, we would need to go through the process of finding the least common multiple of the denominators to proceed.
In our case, matching denominators streamline the subtraction of the numerators: \(x^2 - (5x + 6)\).

Once we have a common denominator, the next step is to focus on the numerators of the fractions. Here, we're looking at \(x^2 - (5x + 6)\). We need to carefully handle the subtraction by applying the distributive property.
Distributing the negative sign, we transform the expression to \(x^2 - 5x - 6\). This step is crucial to ensure accuracy: neglecting the negative sign can lead to errors.
Here’s what happens during this simplification process:

• The negative sign outside the parentheses turns \(5x + 6\) into \(-5x - 6\).


• This changes the original operation from subtracting \(5x + 6\) to adding \(-5x - 6\).

Simplifying the numerator correctly sets the foundation for further operations like polynomial factoring.

Polynomial factoring is the technique of breaking down an expression into simpler parts or factors that, when multiplied, yield the original expression.
In this problem, we factor \(x^2 - 5x - 6\) to simplify further. Since this is a quadratic expression, we look for two numbers that multiply to \(-6\) and add to \(-5\).
These numbers are \(-6\) and \(1\). Factoring the quadratic yields \((x - 6)(x + 1)\).
Factoring helps us simplify expressions drastically:

• It reveals hidden common factors in the numerator and denominator, allowing for cancellation.


• It transforms complex expressions into more manageable ones.

After factoring, we can cancel out the common term \(x - 6\) from both the numerator and denominator, resulting in the simpler form \(x + 1\). This process exemplifies how factoring is key to simplifying algebraic expressions.