How $[4 x^{4} + 8 x^{3} + 8 x^{2} + 4 x + 1]$ can be simplified to $(2 x^{2} + 2 x + 1)^{2}$

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Solution

$4 x^{4} + 8 x^{3} + 8 x^{2} + 4 x + 1 = (2 x^{2})^{2} + 4 x^{2} + 1 + 8 x^{3} + 4 x^{2} + 4 x = (2 x^{2})^{2} + (2 x)^{2} + 1^{2} + 2 (2 x^{2}) (2 x) + 2 (2 x^{2}) (1) + 2 (2 x) (1)$
$= (2 x^{2} + 2 x + 1)^{2}$

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