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It looks like you've uploaded an image of a mathematical expression that you want to solve using the inverse Laplace transform in partial fractions. Let's first break down the expression in the image:

The expression shown is:
\[
\frac{1}{(s-2)(s+3)}
\]

### Steps to solve using Partial Fraction Decomposition:
1. **Express the function as a sum of simpler fractions**:
   \[
   \frac{1}{(s-2)(s+3)} = \frac{A}{s-2} + \frac{B}{s+3}
   \]
   where \(A\) and \(B\) are constants to be determined.

2. **Multiply both sides by \((s-2)(s+3)\)** to get rid of the denominator:
   \[
   1 = A(s + 3) + B(s - 2)
   \]
   Simplify and group like terms:
   \[
   1 = (A + B)s + (3A - 2B)
   \]

3. **Set up a system of equations** by equating coefficients:
   - Coefficient of \(s\): \( A + B = 0 \)
   - Constant term: \( 3A - 2B = 1 \)

4. **Solve the system of equations**:
   - From \( A + B = 0 \), we have \( B = -A \).
   - Substitute into the second equation: \( 3A - 2(-A) = 1 \), which simplifies to \( 3A + 2A = 1 \), so \( 5A = 1 \), hence \( A = \frac{1}{5} \).
   - Since \( B = -A \), \( B = -\frac{1}{5} \).

   So, the partial fraction decomposition is:
   \[
   \frac{1}{(s-2)(s+3)} = \frac{\frac{1}{5}}{s-2} - \frac{\frac{1}{5}}{s+3}
   \]

5. **Apply the inverse Laplace transform**:
   - The inverse Laplace transform of \(\frac{1}{s-a}\) is \( e^{at} \).
   - Therefore:
     \[
     \mathcal{L}^{-1}\left(\frac{1}{(s-2)(s+3)}\right) = \frac{1}{5}e^{2t} - \frac{1}{5}e^{-3t}
     \]

### Final Answer:
\[
\mathcal{L}^{-1}\left(\frac{1}{(s-2)(s+3)}\right) = \frac{1}{5}e^{2t} - \frac{1}{5}e^{-3t}
\]

Would you like any further details or have any questions on this solution? 

Here are some related questions you might consider:
1. How does the Laplace transform of \( \frac{1}{s^2 + a^2} \) compare to this case?
2. What happens if the denominator has repeated roots?
3. How can partial fraction decomposition be applied to higher-degree polynomials?
4. What is the significance of each term in the inverse Laplace transform?
5. How would the procedure change if \( s \) had a complex root?

**Tip**: When solving inverse Laplace problems, always make sure your decomposition is correct by re-multiplying to check your work before applying the inverse transform.

Solve the inverse Laplace transform of 1/(s-2)(s+3) using partial fraction decomposition. This detailed step-by-step solution covers inverse Laplace formulas, partial fraction decomposition, and system of equations.

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