Find the expansion of ${(1 + 2 x - x^{2})}^{4}$ .

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Solution

Using binomial expansion
${(1 + 2 x - x^{2} - ------ -)}^{4}$

$=^{4} C_{0} +^{4} C_{1} (2 x - x^{2}) +^{4} C_{2} {(2 x - x^{2})}^{2} +^{4} C_{3} {(2 x - x^{2})}^{3} +^{4} C_{4} {(2 x - x^{2})}^{4}$

$= 1 + 8 x - 4 x^{2} + 6 (x^{4} - 4 x^{3} + 4 x^{2}) + 4 ({8 x}^{3} - x^{6} + 6 x^{5} - 12 x^{4}) + (16 x^{4} - 32 x^{5} - 8 x^{7} + 24 x^{6} + x^{8}) . .$

$= x^{8} - {8 x}^{7} + {20 x}^{6} - {8 x}^{5} - {26 x}^{4} + {8 x}^{3} + {20 x}^{2} + 8 x + 1$

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