Let's simplify the expression step by step:

$\frac{2a+7}{a^{3}+2a^{2}-a-2}-\frac{3}{a^{2}+a-2}+\frac{2}{a^{2}+3a+2}$

Step 1: Factorize the denominators

We'll start by factoring the quadratic and cubic polynomials in the denominators.

1. Factor $a^{3} + 2a^{2} - a - 2$:

   Let's factor this cubic polynomial by grouping:

   $a^3 + 2a^2 - a - 2 = a^2(a + 2) - 1(a + 2) = (a^2 - 1)(a + 2)$

   Now, notice that $a^2 - 1$ is a difference of squares:

   $a^2 - 1 = (a - 1)(a + 1)$

   So, the factorization of $a^3 + 2a^2 - a - 2$ is:

   $a^3 + 2a^2 - a - 2 = (a - 1)(a + 1)(a + 2)$

2. Factor $a^{2} + a - 2$:

   To factor this quadratic polynomial:

   $a^2 + a - 2 = (a + 2)(a - 1)$

3. Factor $a^{2} + 3a + 2$:

   Factor this quadratic polynomial:

   $a^2 + 3a + 2 = (a + 1)(a + 2)$

Step 2: Rewrite the expression with the factored denominators

Now, rewrite the expression with the factored forms:

$\frac{2a + 7}{(a - 1)(a + 1)(a + 2)} - \frac{3}{(a + 2)(a - 1)} + \frac{2}{(a + 1)(a + 2)}$

Step 3: Combine the fractions

To combine the fractions, identify the least common denominator (LCD), which is $(a - 1)(a + 1)(a + 2)$. Rewrite each fraction with this common denominator:

$\frac{2a + 7}{(a - 1)(a + 1)(a + 2)} - \frac{3(a + 1)}{(a - 1)(a + 1)(a + 2)} + \frac{2(a - 1)}{(a - 1)(a + 1)(a + 2)}$

Step 4: Combine the numerators

Now, combine the numerators over the common denominator:

$\frac{(2a + 7) - 3(a + 1) + 2(a - 1)}{(a - 1)(a + 1)(a + 2)}$

Simplify the numerator:

$2a + 7 - 3a - 3 + 2a - 2 = 2a - 3a + 2a + 7 - 3 - 2 = a + 2$

So, the simplified fraction is:

$\frac{a + 2}{(a - 1)(a + 1)(a + 2)}$

Step 5: Final Simplification

Notice that the numerator $a + 2$ cancels with the factor $a + 2$ in the denominator:

$\frac{1}{(a - 1)(a + 1)}$

Final Answer

The expression simplifies to:

$\boxed{\frac{1}{(a - 1)(a + 1)}}$

Would you like further details or have any questions? Here are five related questions that expand on this topic:

1. How do you approach factoring cubic polynomials?
2. What is the significance of the least common denominator in adding fractions?
3. How do you determine when to cancel out common factors in algebraic fractions?
4. Can you simplify a more complex rational expression with multiple variables?
5. What are the general steps for combining and simplifying rational expressions?

Tip: When dealing with rational expressions, always factorize the denominators first to identify common denominators and simplify the expression.