Give the partial fraction decomposition for the following expression using strategic substitutions for $x$.
$\frac{4x^2-8x-8}{x(x-2)(x+2)}$

• A. $\frac{2}{x}-\frac{1}{x-2}-\frac{3}{x+2}$
• B. $\frac{2}{x}-\frac{1}{x-2}+\frac{3}{x+2}$
• C. $\frac{4}{x^{}}-\frac{8}{x-2}-\frac{8}{x+2}$
• D. $\frac{4}{x^{}}-\frac{8}{x-2}+\frac{8}{x+2}$

Answer: (B)

Verified step by step guidance

1. Step 1: Recognize that the given expression is a rational function, and the goal is to decompose it into partial fractions. The denominator is already factored as \( x(x-2)(x+2) \). This is essential for setting up the partial fraction decomposition.
2. Step 2: Write the general form of the partial fraction decomposition. For \( \frac{4x^2 - 8x - 8}{x(x-2)(x+2)} \), assume \( \frac{4x^2 - 8x - 8}{x(x-2)(x+2)} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2} \), where \( A, B, \) and \( C \) are constants to be determined.
3. Step 3: Multiply through by the denominator \( x(x-2)(x+2) \) to eliminate the fractions. This gives \( 4x^2 - 8x - 8 = A(x-2)(x+2) + Bx(x+2) + Cx(x-2) \). Expand each term on the right-hand side.
4. Step 4: Combine like terms on the right-hand side to match the polynomial on the left-hand side. This will result in a system of equations for \( A, B, \) and \( C \) by equating the coefficients of \( x^2 \), \( x \), and the constant terms.
5. Step 5: Solve the system of equations to find the values of \( A, B, \) and \( C \). Substitute these values back into the partial fraction decomposition \( \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2} \) to complete the decomposition.