Final answer:
To find the form of the partial-fraction decomposition for the given expression, factor the denominator and set up the form with constants for each linear/quadratic term. Solve for these constants to complete the decomposition.
Step-by-step explanation:
The question asks to write the form of the partial-fraction decomposition of the rational expression (x² + 4x - 1)/(x⁴ - 16x²). First, we factor the denominator:
x⁴ - 16x² = x²(x² - 16) = x²(x + 4)(x - 4)
Since the denominator consists of non-repeated linear and quadratic factors, the partial fraction decomposition will be of the form:
A/x² + Bx + C/(x + 4) + Dx + E/(x - 4)
Here, A, B, C, D, and E are constants that will be determined by multiplying both sides of the equation by the original denominator and solving for these constants.