Question 1209875
To solve this problem, we'll use partial fraction decomposition.

**1. Factor the Denominator:**

The denominator of the left side is x³ - 4x. We can factor this as:

x³ - 4x = x(x² - 4) = x(x - 2)(x + 2)

**2. Set up the Partial Fraction Decomposition:**

Now we can write the given expression as a sum of simpler fractions:

(x² - 6x - 3) / \[x(x - 2)(x + 2)] = A/x + B/(x - 2) + C/(x + 2)

where A, B, and C are constants that we need to find.

**3. Clear the Denominators:**

Multiply both sides of the equation by x(x - 2)(x + 2) to get rid of the denominators:

x² - 6x - 3 = A(x - 2)(x + 2) + Bx(x + 2) + Cx(x - 2)

**4. Solve for the Constants:**

We can use various methods to solve for A, B, and C. Here are two common approaches:

   * **Method 1: Substituting Convenient Values of x**

       * **Let x = 0:**
           * 0² - 6(0) - 3 = A(0 - 2)(0 + 2) + B(0)(0 + 2) + C(0)(0 - 2)
           * -3 = A(-2)(2)
           * -3 = -4A
           * A = 3/4

       * **Let x = 2:**
           * 2² - 6(2) - 3 = A(2 - 2)(2 + 2) + B(2)(2 + 2) + C(2)(2 - 2)
           * 4 - 12 - 3 = 0 + B(2)(4) + 0
           * -11 = 8B
           * B = -11/8

       * **Let x = -2:**
           * (-2)² - 6(-2) - 3 = A(-2 - 2)(-2 + 2) + B(-2)(-2 + 2) + C(-2)(-2 - 2)
           * 4 + 12 - 3 = 0 + 0 + C(-2)(-4)
           * 13 = 8C
           * C = 13/8

   * **Method 2: Expanding and Equating Coefficients**

       * Expand the right side of the equation:
           * x² - 6x - 3 = A(x² - 4) + B(x² + 2x) + C(x² - 2x)
           * x² - 6x - 3 = Ax² - 4A + Bx² + 2Bx + Cx² - 2Cx
           * x² - 6x - 3 = (A + B + C)x² + (2B - 2C)x - 4A

       * Equate the coefficients of corresponding powers of x:
           * Coefficient of x²: 1 = A + B + C
           * Coefficient of x: -6 = 2B - 2C
           * Constant term: -3 = -4A

       * Solve the system of equations. From the third equation, A = 3/4. Substituting this into the first two equations and solving yields B = -11/8 and C = 13/8.

**5. Write the Result:**

Substitute the values of A, B, and C back into the partial fraction decomposition:

(x² - 6x - 3) / \[x(x - 2)(x + 2)] = (3/4)/x + (-11/8)/(x - 2) + (13/8)/(x + 2)

So the equation is:

(x² - 6x - 3) / (x³ - 4x) = 3/4 / x + (-11/8) / (x - 2) + 13/8 / (x + 2)

Therefore, the constants are:

* 3/4
* -11/8
* 13/8