A polynomial p$(x)$ leaves the remainder 3 when divided by $(x-1)$ and the remainder 5 when divided by $(x-2)$. The remainder when p$(x)$ is divided by $(x-1)(x-2)$ is:

Dividing by a quadratic leaves a remainder of degree at most 1, say $cx+d$. Writing $p(x)=(x-1)(x-2)q(x)+cx+d$ and putting $x=1$ and $x=2$ gives $c+d=3$ and $2c+d=5$, so $c=2$ and $d=1$ — the remainder is $2x+1$. (Adding the two given remainders to get 8 ignores that the remainder must be a polynomial in $x$.) Check: $2(1)+1=3$ and $2(2)+1=5$.