To decompose the given expression $\frac{2x - 3}{x^2 + x - 6}$, let's proceed with the following steps:

1. Factorize the Denominator: The denominator is $x^2 + x - 6$. We need to factorize this quadratic expression. We look for two numbers that multiply to $-6$ and add up to $1$ (the coefficient of $x$).

   $x^2 + x - 6 = (x + 3)(x - 2)$

   So, the expression becomes:

   $\frac{2x - 3}{(x + 3)(x - 2)}$

2. Attempt Partial Fraction Decomposition: Since the numerator $2x - 3$ has a lower degree than the denominator $(x + 3)(x - 2)$, we can decompose it into partial fractions:

   $\frac{2x - 3}{(x + 3)(x - 2)} = \frac{A}{x + 3} + \frac{B}{x - 2}$

3. Solve for A and B: To find $A$ and $B$, multiply both sides by the denominator $(x + 3)(x - 2)$:

   $2x - 3 = A(x - 2) + B(x + 3)$

   Expand and collect like terms:

   $2x - 3 = Ax - 2A + Bx + 3B$

   Combine terms involving $x$:

   $2x - 3 = (A + B)x + (-2A + 3B)$

   Now, equate the coefficients for $x$ and the constant terms:

   • For $x$: $A + B = 2$
   • For the constant: $-2A + 3B = -3$

4. Solve the System of Equations: From the equations:

   • $A + B = 2$
   • $-2A + 3B = -3$

   Solving these, we get:

   • Multiply the first equation by 2: $2A + 2B = 4$
   • Add this to the second equation: $(2A - 2A) + (2B + 3B) = 4 - 3$

   So, $5B = 1 \Rightarrow B = \frac{1}{5}$.

   Substitute $B = \frac{1}{5}$ into $A + B = 2$:

   $A + \frac{1}{5} = 2 \Rightarrow A = 2 - \frac{1}{5} = \frac{10 - 1}{5} = \frac{9}{5}$

5. Final Decomposition: Substitute $A$ and $B$ back into the partial fractions:

   $\frac{2x - 3}{(x + 3)(x - 2)} = \frac{9}{5(x + 3)} + \frac{1}{5(x - 2)}$

Thus, the decomposed form is:

$\frac{2x - 3}{x^2 + x - 6} = \frac{9}{5(x + 3)} + \frac{1}{5(x - 2)}$

Would you like further details on any step or have additional questions?

Here are some related questions to expand on this topic:

1. How can we perform partial fraction decomposition when the degree of the numerator is higher than the denominator?
2. What are the conditions required for partial fraction decomposition?
3. How do we solve systems of linear equations in general?
4. Why is it useful to decompose rational expressions?
5. How can you check the correctness of a decomposition?

Tip: Always double-check the factoring of the denominator, as any mistakes here will affect the entire decomposition.