1, $x^4+5x^3-12x^2+5x+1$

$=x^4-2x^3+7x^3+x^2-14x^2+x^2+7x-2x+1$

$=\left(x^4-2x^3+x^2\right)+\left(7x^3-14x^2+7x\right)+\left(x^2-2x+1\right)$

$=x^2\left(x^2-2x+1\right)+7x\left(x^2-2x+1\right)+\left(x^2-2x+1\right)$

$=x^2\left(x-1\right)^2+7x\left(x-1\right)^2+\left(x-1\right)^2$

$=\left(x-1\right)^2\left(x^2+7x+1\right)$

2, $6x^4+5x^3-38x^2+5x+6$

$=6x^4+20x^3-15x^3+6x^2-50x^2+6x^2-15x+20x+6$

$=6x^4+20x^3+6x^2-15x^3-50x^2-15x+6x^2+20x+6$

$=\left(2x^2-5x+2\right)\left(3x^2+10x+3\right)$

$=\left(2x^2-x-4x+2\right)\left(3x^2+10x+3\right)$

$=\left[x\left(2x-1\right)-2\left(2x-1\right)\right]\left(3x^2+10x+3\right)$

$=\left(x-2\right)\left(2x-1\right)\left(3x^2+10x+3\right)$

$=\left(x-2\right)\left(2x-1\right)\left(3x^2+9x+x+3\right)$

$=\left(x-2\right)\left(2x-1\right)\left[3x\left(x+3\right)+\left(x+3\right)\right]$

$=\left(x-2\right)\left(2x-1\right)\left(3x+1\right)\left(x+3\right)$