Questions: The middle term of the trinomial has been rewritten. Factor by grouping. 32 t^2 + 36 t + 9 = 32 t^2 + 24 t + 12 t + 9

Transcript text: The middle term of the trinomial has been rewritten. Factor by grouping. \[ 32 t^{2}+36 t+9=32 t^{2}+24 t+12 t+9 \]

Solution

Solution Steps

To factor the trinomial by grouping, we first rewrite the middle term as two separate terms, which has already been done in the problem. Next, we group the terms into two pairs and factor out the greatest common factor from each pair. Finally, we factor out the common binomial factor from the resulting expression.

Step 1: Rewrite the Expression

We start with the trinomial \( 32t^2 + 36t + 9 \). The middle term has been rewritten as \( 32t^2 + 24t + 12t + 9 \).

Step 2: Group the Terms

Next, we group the terms into two pairs: \[ (32t^2 + 24t) + (12t + 9) \]

Step 3: Factor Each Group

Now, we factor out the greatest common factor from each group: \[ 8t(4t + 3) + 3(4t + 3) \]

Step 4: Factor Out the Common Binomial

We notice that \( (4t + 3) \) is a common factor: \[ (4t + 3)(8t + 3) \]