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Rational expressions are fractions where both the numerator and the denominator are polynomials.
These often appear in math problems related to algebra and calculus.

When dealing with rational expressions, it's essential to understand how to manipulate them to solve various math problems, such as simplifying expressions, finding common denominators, or performing partial fraction decomposition.
Partial fraction decomposition involves breaking down a complex rational expression into simpler fractions that are easier to work with.

This technique is beneficial when you're integrating rational functions or solving specific differential equations. The original exercise is a perfect example of using partial fraction decomposition to simplify rational expressions into parts, each with an easily recognizable denominator.

Factorization is a key step in handling rational expressions and understanding partial fraction decomposition.
It involves breaking down a polynomial into simpler polynomials that multiply together to give the original polynomial.

**Difference of Squares**
A common factorization pattern is the difference of squares, expressed as \(a^2 - b^2 = (a-b)(a+b)\). Recognizing and applying this pattern allows you to simplify the original polynomial expression.

For example, in the exercise, the original expression \(x^4 - 16\) is factored as \((x^2 - 4)(x^2 + 4)\), which further factors to \((x-2)(x+2)(x^2+4)\). Each factor is inspected for further factoring possibilities.
Understanding factorization makes it significantly easier to handle complex polynomial expressions and set up the groundwork for decomposition.

Linear equations play a crucial role when breaking down rational expressions into partial fractions.
While factoring, you arrive at expressions with unknown constants such as \(A\), \(B\), \(C\), and \(D\).

**Creating Equations**
The next step involves setting up equations that help find these constants by equating the original numerator to the expression obtained from multiplying each part of the decomposition by the common denominator.
This results in a system where each coefficient of powers of \(x\) in the resulting expressions aligns.

**Example**
In our case, you equate \(6x^2 - 8x + 24\) with \(A(x + 2)(x^2 + 4) + B(x - 2)(x^2 + 4) + (Cx + D)(x - 2)(x + 2)\).
This approach simplifies the process of determining each unknown constant in the decomposition.

Once you set up linear equations for each unknown constant during decomposition, solving them typically involves dealing with systems of equations.
In partial fraction decomposition, especially with multiple unknowns, you encounter a system of linear equations.

**Solving the System**
Solving this system requires aligning coefficients of like terms from both sides of the equation. Each equation corresponds to a specific coefficient's value or a sum of coefficients that equals a particular term in the original polynomial.
This is done to maintain balance and consistency between both sides of the equation.

In the example exercise, solving for \(A\), \(B\), \(C\), and \(D\) yields values by matching coefficients of each term.
Applying methods such as substitution or elimination can help resolve these systems effectively.
Here, using simple algebraic manipulations helps find that \(A=3\), \(B=3\), \(C=0\), and \(D=2\). Recognizing how to efficiently solve these systems can greatly enhance your algebraic problem-solving skills.