(x+a)(x+2a)(x+3a)(x+4a)+a^4
=(x^2+3ax+2a^2)(x^2+7ax+12a^2)+a^4
=(x^4+10ax3+35a2x^2+50a3x+24a^4)+a^4
=x^4+10ax3+35a2x^2+50a^3x+25a^4
=x^2(x^2+10ax+25a^2)+(10a2x^2+50a^3x)+25a^4
=x^2(x+5a)2+10a2x(x+5a)+25a^4
=[x(x+5a)+5a^2]^2