Final answer:
To expand the quotient (x²+4)/(x²-6x+8) using partial fractions, first factor the denominator (x-2)(x-4), then perform long division to express the quotient as a polynomial plus a proper fraction. Next, express the remainder as a sum of partial fractions and solve for the values of the constants.
Step-by-step explanation:
To expand the quotient using partial fractions, we first need to factor the denominator. The denominator, x²-6x+8, can be factored as (x-2)(x-4). Since the degree of the numerator, x²+4, is greater than the degree of the denominator, we start by performing long division to express the quotient as a polynomial plus a proper fraction.
The long division yields a quotient of 1 and a remainder of 6x-12. So the expression can be written as 1 + 6x-12/(x-2)(x-4). Next, we express the remainder as a sum of partial fractions.
The partial fractions can be written as A/(x-2) + B/(x-4), where A and B are constants that we need to find. To find A and B, we multiply both sides of the equation by the denominator and equate coefficients. Solving this system of equations will give us the values of A and B.