Find the polynomial

The fourth degree polynomial equation having one of the roots $1 + \sqrt{2} + 2\sqrt{3}$

is $x^{4} + bx^{3} + cx^{2} + dx + e = 0$

Find 1 + b + c + d + e ( all are integers)

Details - Question appeared in Ramanujan Ganit Spardha 2014 Std XI and XII (Only Gujarat region)

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This exam motive is to just know how differently you can approach a basic question(and if we have a different and short method we are rewarded extra marks). So please write your method if you got your answer right.