Final answer:
The student's question involves expressing quotients into partial fractions: (1)/(4-x²) and (x²)/((x-1)(x²+2x+1)). The process requires factoring the denominators and setting up sums with unknown coefficients, which are then found by equating coefficients and solving for the variables.
Step-by-step explanation:
The question deals with expressing given quotients into partial fractions. For the first expression (i) (1)/(4-x²), we recognize that 4 - x² is the difference of squares and can be factored into (2+x)(2-x). Thus, we can rewrite the fraction as a sum of partial fractions in the form A/(2+x) + B/(2-x).
For the second expression (ii) (x²)/((x-1)(x²+2x+1)), we notice that the denominator can be further factored since x²+2x+1 is a perfect square (x+1)². This means we can express this fraction as a sum of the form C/(x-1) + (Dx + E)/(x+1)².
To find the values of A, B, C, D, and E, we equate coefficients after multiplying both sides by the common denominator and solving the resulting system of linear equations. These coefficients allow us to complete the partial fraction decomposition for each expression.