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Quadratic expressions come in the form of \(\textstyle{x^2 + bx + c}\). Factoring these is essential when working with rational expressions.
First, identify two numbers that multiply to \(\textstyle{c}\) (the constant term) and add up to \(\textstyle{b}\) (the coefficient of the linear term). For the expression \(\textstyle{t^2 - 5t + 6}\), you need numbers that multiply to \(\textstyle{6}\) and add to \(\textstyle{-5}\).
These numbers are \(\textstyle{-2 \text{ and } -3}\). Therefore, \(\textstyle{t^2 - 5t + 6}\) can be factored into \(\textstyle{(t-3)(t-2)}\).
Breaking down these quadratic expressions into products of binomials simplifies the process of multiplying and canceling terms in rational expressions.

Simplifying fractions involves reducing them to their simplest form by canceling common factors between the numerator and the denominator.
When fractions are multiplied, you can first rewrite one large fraction by combining the numerators and denominators. In this example:
\(\frac{t+2}{t-2} \times \frac{(t-3)(t-2)}{(t+2)^2}\rightarrow\frac{(t+2)(t-3)(t-2)}{(t-2)(t+2)^2}\), combining the two fractions into one.
This makes it easier to spot and cancel common factors.
Start by listing out all the factors in the numerator and the denominator, looking for any matches. For successful simplification, it's critical to factor both the numerator and the denominator completely first.
Having common factors in both parts allows you to simplify the fraction effectively.

Canceling common factors is straightforward once the expression is properly factorized.
Look for identical factors in both the numerator and the denominator. In our example:
\(\frac{(t+2)(t-3)(t-2)}{(t-2)(t+2)^2}\), note that both \(\textstyle{t-2}\) can be canceled from the numerator and the denominator.
Similarly, one \(\textstyle{t+2}\) appears in both parts, allowing you to cancel them as well.
This reduces the fraction to: \(\frac{t-3}{t+2}\).
Remember, canceling common factors simplifies the fraction, making it easier to work with. Be sure to only cancel terms that are multiplied, not added or subtracted, to maintain the equality of the expression. Proper canceling is key for accurate simplifying.