The given mathematical expression is:

$\frac{4x^2 - 9}{2x^3 - 3x^2 - 8x + 12}.$

This is a rational function that can be simplified if possible. Let me start by simplifying the numerator and denominator.

Step 1: Factorize the numerator $4x^2 - 9$

This is a difference of squares: $4x^2 - 9 = (2x - 3)(2x + 3).$

Step 2: Factorize the denominator $2x^3 - 3x^2 - 8x + 12$

Group terms to simplify factorization: $2x^3 - 3x^2 - 8x + 12 = (2x^3 - 3x^2) + (-8x + 12).$ Factorize each group: $= x^2(2x - 3) - 4(2x - 3).$ Now, factor out the common term $(2x - 3)$: $= (2x - 3)(x^2 - 4).$

Further factorize $x^2 - 4$ as it is a difference of squares: $x^2 - 4 = (x - 2)(x + 2).$

Thus, the denominator becomes: $2x^3 - 3x^2 - 8x + 12 = (2x - 3)(x - 2)(x + 2).$

Step 3: Simplify the rational function

The expression is now: $\frac{(2x - 3)(2x + 3)}{(2x - 3)(x - 2)(x + 2)}.$

Cancel out the common factor $(2x - 3)$: $\frac{2x + 3}{(x - 2)(x + 2)}.$

Final Answer:

The simplified form of the rational function is: $\frac{2x + 3}{(x - 2)(x + 2)}.$

Let me know if you'd like further details or have any questions!

Related Questions:

1. How do you determine when to cancel terms in a rational function?
2. What is the significance of factoring in simplifying rational expressions?
3. How does the difference of squares help in factorization?
4. What are the restrictions on $x$ for the given function (values that make the denominator zero)?
5. Can this simplified expression be used for graphing the rational function?

Tip:

Always check for any restrictions in the domain of a rational function by identifying the roots of the denominator before simplifying!