To simplify the function \( f(x) = 3 - 5(6 - t) - 2(6 - t)^2 \), let's follow these steps:

1. Expand the squared term \((6 - t)^2\).
2. Distribute the constants \(-5\) and \(-2\).
3. Combine like terms.

### Step-by-Step Solution:

1. **Expand \((6 - t)^2\):**
   \[
   (6 - t)^2 = 6^2 - 2 \cdot 6 \cdot t + t^2 = 36 - 12t + t^2
   \]

2. **Distribute the constants:**

   For the term \(-5(6 - t)\):
   \[
   -5(6 - t) = -5 \cdot 6 + 5 \cdot t = -30 + 5t
   \]

   For the term \(-2(36 - 12t + t^2)\):
   \[
   -2(36 - 12t + t^2) = -2 \cdot 36 + 2 \cdot 12t - 2 \cdot t^2 = -72 + 24t - 2t^2
   \]

3. **Combine all terms together:**

   \[
   f(t) = 3 - 30 + 5t - 72 + 24t - 2t^2
   \]

   Simplify by combining like terms:
   \[
   f(t) = (3 - 30 - 72) + (5t + 24t) - 2t^2
   \]

   \[
   f(t) = -99 + 29t - 2t^2
   \]

Therefore, the simplified function is:
\[
f(t) = -2t^2 + 29t - 99
\]

Would you like further details on any part of this process, or do you have any questions?

### Relative Questions:

1. What are the steps to simplify a polynomial expression?
2. How do you expand a binomial squared?
3. What is the process for distributing constants over terms in parentheses?
4. How do you combine like terms in a polynomial?
5. What is the importance of arranging terms in standard form?
6. How do you factor a quadratic expression?
7. What are the methods for finding the roots of a quadratic equation?
8. How can the derivative of \( f(t) \) be calculated?

### Tip:
When simplifying complex expressions, always work step-by-step and combine like terms carefully to avoid mistakes.

Learn how to simplify the polynomial function f(x) = 3 - 5(6 - t) - 2(6 - t)^2 step-by-step. This problem covers polynomial simplification, binomial expansion, and combining like terms. Suitable for high school students.

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