{ (s+2) / (s^3 - 5s^2 + 4s) } Find laplace

To find the Laplace transform of the function 

\[
\frac{s+2}{s^3 - 5s^2 + 4s}
\]

we can start by simplifying the expression by factoring the denominator.

### Step 1: Factor the Denominator
The denominator \( s^3 - 5s^2 + 4s \) can be factored as:

\[
s^3 - 5s^2 + 4s = s(s^2 - 5s + 4) = s(s - 1)(s - 4)
\]

So, the expression becomes:

\[
\frac{s+2}{s(s - 1)(s - 4)}
\]

### Step 2: Partial Fraction Decomposition
We can decompose this fraction into partial fractions:

\[
\frac{s+2}{s(s - 1)(s - 4)} = \frac{A}{s} + \frac{B}{s - 1} + \frac{C}{s - 4}
\]

To find \( A \), \( B \), and \( C \), multiply both sides by \( s(s - 1)(s - 4) \) to eliminate the denominators:

\[
s + 2 = A(s - 1)(s - 4) + B(s)(s - 4) + C(s)(s - 1)
\]

Now we expand and solve for \( A \), \( B \), and \( C \).

1. Let \( s = 0 \):  
   \[
   0 + 2 = A(0 - 1)(0 - 4) \Rightarrow

Discover how to find the Laplace transform of the function (s+2) / (s^3 - 5s^2 + 4s) through step-by-step factorization and partial fraction decomposition. This problem is designed for students in Grades 11-12 studying advanced mathematics.

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