jmc

We begin by expanding $(a+10)(b+10)$ out: $$\begin{array}{rl}(a+10)(b+10)& =a(b+10)+10(b+10)\\ & =ab+10a+10b+100\\ & =ab+10(a+b)+100.\end{array}$$This is congruent modulo $20$ to $ab+10(a+b)$.

Now we observe that $ab$ is $17$ more than a multiple of $20$, so $ab$ is odd, which means $a$ and $b$ must each be odd. Therefore, $a+b$ is even, so $10(a+b)$ is a multiple of $20$. It follows that $$ab+10(a+b)\equiv ab\equiv \boxed{17}(\mathrm{mod}20).$$