The addition of strain and p-doping in high-speed GaAs-based MQW lasers are assessed experimentally in terms of their potential for both increasing the differential gain, partial derivativeg/partial derivativeN, and reducing the modulus of the differential refractive index, \partial derivativen/partial derivativeN\, in order to decrease the linewidth enhancement factor, alpha. The increased differential gain with strain alone is found to be offset by a corresponding increase of \partial derivativen/partial derivativeN\. The further addition of p-doping, on the other hand, simultaneously increases partial derivativeg/partial derivativeN and decreases \partial derivativen/partial derivativeN\, yielding a substantial reduction in alpha.