Remainder!

let $P(x)=x^3 -x^2, F(x)=x^{81}+x^{49}+x^{25}+x^9+x$ and $r(x) =remainder$ than $F(x)=P(x)Q(x)+r(x)$ for some polynomial Q(x) $F(x)-r(x)=P(x)Q(x)$ and hence $P(x)|F(x)-r(x)$ that also means that if $P(a_1)=P(a_2)=0, F(a_1)-r(a_1)=F(a_2)-r(a_2)=0$ we see that the roots of $P(x)=0, x =(1,0)$ hence $F(1)-r(1)=0\longrightarrow r(1)=5$ and $F(0)-r(0)=0\longrightarrow r(0)=0$ since zero is a root of r(x) and the its maximum degree is 2, it can be written as $r(x)=(x+0)(zx+n)$ from 1 of the eqs, $r(1)=5=1(z+n)--> z+n=5$ we see that the only choice from the option satisfying this is $\boxed{4x^2+x}$

Substitute 1 into both equations. Observe the remainder is 5. Therefore the only answer is 4xsqaured +x

at x=1 the GE=5 and also 4x^2+x=5 at x=1 simple!