$x^{11} + x^{4} + 1$
$= (x^{11} + x^{10} + x^{9}) - (x^{10} + x^{9} + x^{8}) + (x^{8} + x^{7} + x^{6}) - (x^{7} + x^{6} + x^{5}) + (x^{5} + x^{4} + x^{3}) - (x^{3} + x^{2} + x) + (x^{2} + x + 1)$
$= x^{9}(x^{2} + x + 1) - x^{8}(x^{2} + x + 1) + x^{6}(x^{2} + x + 1) - x^{5}(x^{2} + x + 1) + x^{3}(x^{2} + x + 1) - x(x^{2} + x + 1) + (x^{2} + x + 1)$
$= (x^{2} + x + 1)(x^{9} - x^{8} + x^{6} - x^{5} + x^{3} - x + 1)$