Question

factor completely.
$3t^2 - 66t - 69$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of the terms \(3t^2\), \(-66t\), and \(-69\). The GCF of 3, 66, and 69 is 3. Factor out 3 from each term:
\(3t^2 - 66t - 69 = 3(t^2 - 22t - 23)\)

Step2: Factor the quadratic trinomial

Now, factor the quadratic \(t^2 - 22t - 23\). We need two numbers that multiply to \(-23\) and add up to \(-22\). The numbers are \(-23\) and \(1\) because \(-23 \times 1 = -23\) and \(-23 + 1 = -22\). So we can factor the quadratic as:
\(t^2 - 22t - 23 = (t - 23)(t + 1)\)

Step3: Combine the factors

Putting it all together, the completely factored form of the original expression is:
\(3(t - 23)(t + 1)\)

Answer:

\(3(t - 23)(t + 1)\)