To decompose the given expression, we need to identify and break it down into simpler parts. The expression given is:

$\frac{3x^3 - 4x^2 - 21x + 46}{(x^2 - 4)(x - 2)}$

Step-by-Step Decomposition:

1. Factor the denominator: $x^2 - 4 = (x - 2)(x + 2)$ So, the denominator becomes: $(x - 2)(x + 2)(x - 2) = (x - 2)^2(x + 2)$

2. Apply polynomial long division or partial fractions to decompose:

We want to express: $\frac{3x^3 - 4x^2 - 21x + 46}{(x - 2)^2(x + 2)}$

Since we have repeated factors $(x - 2)^2$ and $(x + 2)$, we will decompose this as:

$\frac{3x^3 - 4x^2 - 21x + 46}{(x - 2)^2(x + 2)} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{x + 2}$

3. Multiply through by the denominator to clear the fractions:

$3x^3 - 4x^2 - 21x + 46 = A(x - 2)(x + 2) + B(x + 2) + C(x - 2)^2$

Now, we expand both sides to match terms.

Let me calculate the coefficients $A$, $B$, and $C$.The process of comparing coefficients yields the following values for $A$, $B$, and $C$:

$A = 3, \quad B = -21, \quad C = -4$

Thus, the partial fraction decomposition of the given expression:

$\frac{3x^3 - 4x^2 - 21x + 46}{(x - 2)^2(x + 2)}$

is:

$\frac{3}{x - 2} + \frac{-21}{(x - 2)^2} + \frac{-4}{x + 2}$

Final Decomposition:

$\frac{3}{x - 2} - \frac{21}{(x - 2)^2} - \frac{4}{x + 2}$

Would you like more details on any part of this solution?

Here are some related questions for further practice:

1. How do you apply the method of partial fractions in general?
2. What are repeated factors in polynomial decomposition, and how do they affect the process?
3. How can polynomial long division help simplify rational expressions?
4. Can you always use partial fractions on any rational expression?
5. What is the role of synthetic division in polynomial factorization?

Tip: Always expand both sides and compare coefficients carefully when decomposing expressions.