First notice that adding or substracing two interesting numbers you get another interesting number. And if we let $r=a+b\sqrt2$ and $s=c+d\sqrt2$ be two interesting numbers, by multiplying them $r\cdot s=(a+b\sqrt2)(c+d\sqrt2)=(ac+2bd)+(ad+bc)\sqrt2$ we get another interesting number. So let us denote the polynominals as $A(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ $B(x)=b_mx^m+b_{m-1}x^{m-1}+...b_1x+1$ $Q(x)=c_kx^k+c_{k-1}x^{k-1}+...+c_1x+c_0$ So that $n,m,k\in N$ with $n=m\cdot k$ and $\{a_i\}_0^n$ with $\{b_i\}_1^m$ are interesting coefficients and where $\{c_i\}_0^k$ are real numbers. We prove the claim by strong induction on $\{c_i\}_0^k$. Multiplying and equaling constand terms in $A(x)=B(x)\cdot Q(x)$ we get $c_0=a_0$. So $c_0$ is interesting and if the terms of $Q(x)$ up to $i-1$ are interesting, for $i$ we have $c_ix^i+tx^i=a_ix^i$ $c_i=a_i-t$ Where $t$ is stuff from multiplying previous interesting terms of $A(x),B(x)$ and $Q(x)$ so it is interesting as well and therefore so is $c_i$.