2015/012) find $12x^4-2x^3-25x^2+ 9x + 2017$ given $x= \dfrac{\sqrt{5}+1}{4}$

we have $4x-1 = \sqrt5$

squaring and reordering we get
$16x^2-8x -4-0$
or $4x^2-2x-1 = 0 \cdots (1)$

now deviding $12x^4-2x^3-25x^2+ 9x + 2017$ by $(4x^2-2x-1)$ we find that

$12x^4-2x^3-25x^2+9x+2017$
$=(4x^2-2x-1)(3x^2+x+5) + 2012= 2012$ using (1)