The LCM and HCF of two polynomials p(x) and q(x) are (x + a)(x³ – a³) and (x² – ax + a²), respectively. If p(x) = x⁴ + a²x² + a⁴, then what is q(x) equal to?

Using the relation p(x) × q(x) = LCM × HCF. LCM = (x+a)(x³ – a³) = (x+a)(x–a)(x²+ax+a²) = (x²–a²)(x²+ax+a²). HCF = x² – ax + a². So LCM × HCF = (x²–a²)(x²+ax+a²)(x²–ax+a²) = (x²–a²)(x⁴+a²x²+a⁴). Given p(x) = x⁴ + a²x² + a⁴, so q(x) = LCM × HCF / p(x) = (x²–a²)(x⁴+a²x²+a⁴)/(x⁴+a²x²+a⁴) = x² – a².