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Let's break down and solve the given expression step by step.

The expression is:

\[
(2t - 3)(3t + 2) + 6t - 3
\]

It equals \( At^2 + Bt + C \), where we need to find \( A \), \( B \), and \( C \).

### Step 1: Expand the expression \( (2t - 3)(3t + 2) \)

Using the distributive property (FOIL method):

\[
(2t - 3)(3t + 2) = 2t(3t) + 2t(2) - 3(3t) - 3(2)
\]

Simplifying each term:

\[
= 6t^2 + 4t - 9t - 6
\]

\[
= 6t^2 - 5t - 6
\]

### Step 2: Add the remaining terms

Now, we need to add \( 6t - 3 \) to the expanded expression:

\[
6t^2 - 5t - 6 + 6t - 3
\]

Simplify the like terms:

\[
= 6t^2 + (-5t + 6t) + (-6 - 3)
\]

\[
= 6t^2 + t - 9
\]

### Step 3: Compare with \( At^2 + Bt + C \)

By comparing:

\[
6t^2 + t - 9 = At^2 + Bt + C
\]

We can see that:
- \( A = 6 \)
- \( B = 1 \)
- \( C = -9 \)

### Final Answer:
- \( A = 6 \)
- \( B = 1 \)
- \( C = -9 \)

Let me know if you want further details on any of these steps.

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Here are some related questions to expand your understanding:

1. How would the result change if \( 6t - 3 \) was replaced with \( 5t - 2 \)?
2. What are the key steps in using the distributive property to expand polynomials?
3. How does combining like terms affect the final result in such equations?
4. How can you verify your result by factoring the final quadratic expression?
5. What would be the effect of increasing the coefficients in the original binomials?

**Tip:** Always double-check your expanded terms and ensure like terms are combined correctly for accurate results!

The expression (2t - 3)(3t + 2) + 6t - 3 equals At^2 + Bt + C

Learn how to solve the expression (2t - 3)(3t + 2) + 6t - 3 step-by-step. This problem covers key algebraic concepts, including expanding polynomials and combining like terms. Suitable for students in Grades 7-9.

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