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Factoring is the process of breaking down a complex expression, like a quadratic, into simpler components or products. For quadratics, this typically involves rewriting the expression as a product of two binomials. This is often helpful in calculus, particularly when evaluating limits. For example, consider the quadratic expression in the numerator, \(t^{2} + 6t + 5\). We factor it into \((t + 1)(t + 5)\). Similarly for the denominator \(t^{2} + 3t + 2\), it can be factored into \((t + 1)(t + 2)\).

Factoring provides a way to simplify expressions, making them easier to manipulate in later steps, such as canceling common terms.

Evaluating limits is a fundamental concept in calculus. It involves finding the value that a function approaches as the input approaches a certain point. In the given exercise, we want to find the limit of \(\frac{t^{2}+6t+5}{t^{2}+3t+2}\) as \(t\) approaches \(-1\). After factoring as discussed, we simplified the expression to \(\frac{(t + 1)(t + 5)}{(t + 1)(t + 2)}\).

Once simplification is done, we can proceed to evaluate the limit by substituting the value of \(t\) that it approaches, here \(-1\). If the direct substitution yields a determinate form, that result could be the limit. In this case, simplifying further is crucial to avoiding an indeterminate form caused by division by zero.

Canceling common factors is an essential step when handling limits involving fractions. It helps avoid situations like division by zero, which can cause complications. When you have a common factor in both the numerator and denominator, like \((t + 1)\) in our example, you can 'cancel' this factor. This doesn't affect the expression except where the factor equals zero.

After canceling \((t + 1)\), the limit simplifies to \(\frac{t + 5}{t + 2}\). This step remains key, especially when dealing with limits that result in indeterminate forms such as \(\frac{0}{0}\), a signal that there's more simplification needed, often through cancelation.

Completing this last step allows you to evaluate the limit by substitution without issues, leading easily to the final result.