$a)$ $2x^4 - 21x^3 - 30x^2 - 105x + 50$

$= 2x^4 - 25x^3 + 4x^3 + 10x^2 - 50x^2 + 10x^2 + 20x - 125x + 50$

$= (2x^4 - 25x^3 + 10x^2) + (4x^3 - 50x^2 + 20x) + (10x^2 - 125x + 50)$

$= x^2(2x^2 - 25x + 10) + 2x(2x^2 - 25x + 10) + 5(2x^2 - 25x + 10)$

$= (2x^2 - 25x + 10)(x^2 + 2x + 5)$

$c)$ $x^4 + 6x^3 + 7x^2 - 6x + 1$

$= x^4 + 6x^3 - 2x^2 + 9x^2 - 6x + 1$

$= (x^4 + 6x^3 - 2x^2) + (3x - 1)^2$

$= x^2(x^2 + 6x - 2) + (3x - 1)^2$

$= x^4 + 3x^3 - x^2 + 3x^3 + 9x^2 - 3x - x^2 - 3x + 1$

$= x^2(x^2 + 3x - 1) + 3x(x^2 + 3x - 1) - 1(x^2 + 3x - 1)$

$= (x^2 + 3x - 1)(x^2 + 3x - 1)$

$= (x^2 + 3x - 1)^2$

$e)$ $x^4 + 10x^3 + 26x^2 + 10x + 1$

$= x^4 + 4x^3 + x^2 + 6x^3 + 24x^2 + 6x + x^2 + 4x + 1$

$= x^2(x^2 + 4x + 1) + 6x(x^2 + 4x + 1) + 1(x^2 + 4x +1)$

$= (x^2 + 4x + 1)(x^2 + 6x + 1)$

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