To solve the given expression [tex]\(( -x^2 + 2x + 4)^{32}\)[/tex], we need to expand it. This expansion would typically involve the binomial theorem applied repeatedly, which can be arduous due to the high power (32) and the polynomial form. For simplicity, let's directly present the result of this expansion.
The expanded form of [tex]\(( -x^2 + 2x + 4)^{32}\)[/tex] is:
[tex]\[ x^{64} - 64x^{63} + 1856x^{62} - 31744x^{61} + 345216x^{60} - 2317312x^{59} + 7047168x^{58} + 25776128x^{57} - 365468672x^{56} + 1296171008x^{55} + 2224111616x^{54} - 34819866624x^{53} + 81835524096x^{52} + 316030058496x^{51} - 2073637945344x^{50} + 678722273280x^{49} + 24478189486080x^{48} - 53901860536320x^{47} - 171264253624320x^{46} + 785734151700480x^{45} + 554176736133120x^{44} - 7368208998727680x^{43} + 2934795109662720x^{42} + 52549393286430720x^{41} - 59955610732462080x^{40} - 306352429090209792x^{39} + 533916604786802688x^{38} + 1532739988171522048x^{37} - 3445811564618186752x^{36} - 6871690542497923072x^{35} + 17914233618666881024x^{34} + 28721927144356184064x^{33} - 77930613325856505856x^{32} - 114887708577424736256x^{31} + 286627737898670096384x^{30} + 439788194719867076608x^{29} - 882127760542255808512x^{28} - 1569525747887638577152x^{27} + 2186922413206743810048x^{26} + 5019278198213997232128x^{25} - 3929250904962634874880x^{24} - 13775508153678094663680x^{23} + 3077355716909696286720x^{22} + 30904508476199503134720x^{21} + 9297542804280358993920x^{20} - 52729726326622881054720x^{19} - 45973398018143991889920x^{18} + 57876682049261855047680x^{17} + 105133023308004647239680x^{16} - 11660359867217498603520x^{15} - 142499314543953847517184x^{14} - 86869681010770364596224x^{13} + 89979110308695030890496x^{12} + 153139392922789814992896x^{11} + 39126985332218717536256x^{10} - 91209926072456877965312x^{9} - 102870285939674376568832x^{8} - 29021340113963552079872x^{7} + 31737623178817283555328x^{6} + 41744981838804715307008x^{5} + 24875434383397330354176x^{4} + 9149585060559937601536x^{3} + 2139822312550307987456x^{2} + 295147905179352825856x + 18446744073709551616 \][/tex]
Therefore, the expanded form of the given expression is as stated above. This final expanded form represents a polynomial in [tex]\(x\)[/tex] with coefficients that result from the binomial expansions and collecting like terms.
