https://file.aiquickdraw.com/mathbot/file/1725935155569rk7bgwej.jpg

The expression in the image is:

\[
-\left( \frac{t^2}{3} + 1 \right)\left( \frac{1}{2} - \frac{t^2}{6} \right)
\]

To expand this, we will use the distributive property (also known as the FOIL method for binomials) and distribute the terms inside both parentheses.

Step-by-step:

1. First, distribute \(-\left( \frac{t^2}{3} \right)\) to each term in the second parenthesis:

\[
-\left( \frac{t^2}{3} \right) \times \frac{1}{2} = -\frac{t^2}{6}
\]
\[
-\left( \frac{t^2}{3} \right) \times -\frac{t^2}{6} = +\frac{t^4}{18}
\]

2. Next, distribute \(-1\) to each term in the second parenthesis:

\[
-1 \times \frac{1}{2} = -\frac{1}{2}
\]
\[
-1 \times -\frac{t^2}{6} = +\frac{t^2}{6}
\]

Now, combining all the terms:

\[
-\frac{t^2}{6} + \frac{t^4}{18} - \frac{1}{2} + \frac{t^2}{6}
\]

Notice that \(-\frac{t^2}{6} + \frac{t^2}{6}\) cancels out, leaving:

\[
\frac{t^4}{18} - \frac{1}{2}
\]

Thus, the expanded form of the expression is:

\[
\frac{t^4}{18} - \frac{1}{2}
\]

Let me know if you'd like any additional details or have further questions!

Here are 5 related questions to explore:
1. How can the distributive property be applied to other types of expressions?
2. What are the conditions for binomial expansion when the terms are more complex?
3. Can this method be generalized to polynomials of higher degree?
4. How does the order of operations affect the expansion of expressions like this?
5. How do negative signs in front of parentheses impact multiplication and expansion?

**Tip**: Always ensure to simplify like terms during expansion to avoid unnecessary complexity.

Expand and simplify the algebraic expression -(t^2/3 + 1)(1/2 - t^2/6) using the distributive property. Learn step-by-step how to distribute and combine like terms, with solutions suited for Grades 9-12.

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