(x-6)(x-3)(x+2)(x+4)-84x^2
=((x-6)(x+2))((x-3)(x+4))-84x^2
=(x^2-4x-12)(x^2+x-12)-84x^2
=(x^2-12)^2-3x(x^2-16)-4x^2-84x^2
=(x^2-12)^2-3x(x^2-16)-88x^2
=(x^2-12-11x)(x^2+8x-12)
=(x-12)(x+1)(x^2+8x-12)
(x+a)(x+2a)(x+3a)(x+4a)+a^4
=((x+a)(x+4a))((x+2a)(x+3a))+a^4
=(x^2+5ax+4a^2)(x^2+5ax+6a^2)+a^4
=(x^2+5ax)^2+10a^2(x^2+5ax)+24a^2+a^4
=(x^2+5ax)^2+10a^2(x^2+5ax)+25a^2
=(x^2+5ax+5a^2)