Let's simplify the expression step by step:

\[
\left( \frac{3t}{t^2} - 25 \right) - \left( \frac{t+1}{t^2} + 5t \right)
\]

### Step 1: Distribute and combine like terms.

The expression is:

\[
\frac{3t}{t^2} - 25 - \left( \frac{t+1}{t^2} + 5t \right)
\]

First, distribute the negative sign:

\[
\frac{3t}{t^2} - 25 - \frac{t+1}{t^2} - 5t
\]

### Step 2: Simplify the fractions.

We can simplify \(\frac{3t}{t^2}\) and \(\frac{t+1}{t^2}\):

\[
\frac{3t}{t^2} = \frac{3}{t}, \quad \frac{t+1}{t^2} = \frac{1}{t} + \frac{1}{t^2}
\]

Now substitute back into the expression:

\[
\frac{3}{t} - 25 - \left( \frac{1}{t} + \frac{1}{t^2} + 5t \right)
\]

### Step 3: Combine like terms.

Now, subtract the fractions and other terms:

\[
\left( \frac{3}{t} - \frac{1}{t} \right) - 25 - \frac{1}{t^2} - 5t
\]

Simplify the terms with \(\frac{3}{t} - \frac{1}{t}\):

\[
\frac{2}{t} - 25 - \frac{1}{t^2} - 5t
\]

### Final simplified expression:

\[
\frac{2}{t} - 25 - \frac{1}{t^2} - 5t
\]

This is the simplified form of the expression.

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Do you want further details or have any questions? Here are 5 related questions to explore:

1. How do you combine fractions with different denominators?
2. What happens when \( t = 0 \) in the simplified expression?
3. Can you factor the final expression further?
4. How does the expression behave for large values of \( t \)?
5. What are the vertical and horizontal asymptotes of this expression?

**Tip:** When simplifying fractions, always check for common factors or terms that can be combined early on to streamline your work!

Step-by-step solution for simplifying the rational expression ((3t)/(t^2) - 25) - ((t+1)/(t^2) + 5t). Learn how to distribute, combine like terms, and simplify fractions. Ideal for students in Grades 10-12 learning advanced algebra concepts.

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