https://file.aiquickdraw.com/mathbot/file/172635215770396vebx8g.jpg

The image shows a mathematical expression related to the **Laplace Transform**. The expression reads:

\[
\frac{s^2 + 2s + 2}{s^2 + 7s}
\]

If you're looking to apply the **Laplace Transform**, or to simplify or factor this expression, here's how we could approach it:

1. **Factor the denominator**: The denominator \( s^2 + 7s \) can be factored as \( s(s + 7) \).

   \[
   \frac{s^2 + 2s + 2}{s(s + 7)}
   \]

2. **Partial Fraction Decomposition**: If we want to decompose this into simpler fractions (as often done in Laplace Transform problems), we express the fraction as a sum of simpler terms:
   
   \[
   \frac{s^2 + 2s + 2}{s(s+7)} = \frac{A}{s} + \frac{B}{s+7}
   \]

   To find the values of \( A \) and \( B \), we multiply both sides by the common denominator \( s(s + 7) \) and then solve for \( A \) and \( B \) by comparing coefficients. However, before proceeding, it is important to confirm the exact goal. 

Would you like to proceed with the partial fraction decomposition, or do you need the Laplace Transform of a related function? 

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### Possible Related Questions:
1. How do you solve for the constants in partial fraction decomposition?
2. What are the common Laplace Transforms of rational functions?
3. How do you compute the inverse Laplace Transform of a fraction?
4. What role do poles play in solving Laplace Transforms?
5. Can this expression be simplified further?

#### Tip:
When simplifying Laplace Transform expressions, factoring polynomials in the denominator and applying partial fractions help in breaking down complex functions into simpler components.

Learn how to apply the Laplace Transform to an algebraic fraction and simplify using partial fraction decomposition. This example uses the expression (s^2 + 2s + 2) / (s^2 + 7s), showing step-by-step how to factor polynomials and solve for constants in partial fractions.

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