$(x+1)(x+3)(x+5)(x+7)+15$
$ = [(x+1)(x+3)][(x+3)(x+5)] + 15$
$ = (x^{2}+8x+7)(x^{2}+8x+15)+15$
$ = (x^{2} + 8x -11 - 4)(x^{2}+8x+11+4) + 15$
$ = (x^{2}+8x+11)^{2} -16 + 15$
$ = (x^{2}+8x+11)^{2} -1$
$ = (x^{2} + 8x +11-1)(x^{2}+8x+11+1)$
$ = (x^{2}+8x+10)(x^{2}+8x+12)$