Solve the equation $x^4-8x^3+23x^2-30x+15=0$
Solution 1:
Note that, if $p(x)$ is your polynomial, then $p(x+2)=x^4-x^2-2x-1=x^4-(x+1)^2$. Can you take it from here?
Solution 2:
Since you tried the rational root theorem and didn't find any rational roots, then next thing to try is factoring as a product of two quadratics. Set your polynomial equal to
$$(x^2+ax+b)(x^2+cx+d)$$ $$=x^4 + (a+c)x^3+(d+ac+b)x^2+(ad+bc)x+bd$$
and equate coefficients. Since $bd=15$ you can try $b=3$, $d=5$ and see what $a$ and $c$ turn out to be. If that doesn't work, try $b=1$, $d=15$.