Your are given an equation $$ ax^2+(b+c)x+d+e=0 $$ whose roots are real and greater than $1,$ show that the equation $$ ax^4+bx^3+cx^2+dx+e $$ has at least one real root. Note that $a,b,c,d,e$ are all real numbers.
I have tried to solve this question using the method of contradiction. However I was not able to make great progress. Is there any way I can solve it using the method of contradiction or any other proof strategy. I would also love to see an approach using predicate logic (only if it's possible).