An element does not belong to an ideal 
How can I prove that the element $x-5$ does not belong to the ideal $(x^2-25,-4x+20)$ in $\mathbb Z[x]$. 

I tried to show that by proving $x-5\neq(x^2-25)f(x)+(-4x+20)g(x)$ for all $f,g$. Any help?

${\rm mod}\ \color{#c00}2\!:\  (x^2\!-25) f -\color{#c00}2\, g\ $ is either $\,0\,$ or of degree $\ge 2,\,$ so is $\,\not\equiv\, x-5$
