$A=\frac{\sqrt{3-2\sqrt{2}}}{\sqrt{17-12\sqrt{2}}}-\frac{\sqrt{3+2\sqrt{2}}}{\sqrt{17+12\sqrt{2}}}$$=\frac{\sqrt{{{\left( \sqrt{2}-1 \right)}^{2}}}}{\sqrt{{{\left( 3-2\sqrt{2} \right)}^{2}}}}-\frac{\sqrt{{{\left( \sqrt{2}+1 \right)}^{2}}}}{\sqrt{{{\left( 3+2\sqrt{2} \right)}^{2}}}}$
$=\frac{\sqrt{2}-1}{3-2\sqrt{2}}-\frac{\sqrt{2}+1}{3+2\sqrt{2}}$$=\frac{\sqrt{2}-1}{{{\left( \sqrt{2}-1 \right)}^{2}}}-\frac{\sqrt{2}+1}{{{\left( \sqrt{2}+1 \right)}^{2}}}$$=\frac{1}{\sqrt{2}-1}-\frac{1}{\sqrt{2}+1}$
$=\frac{\sqrt{2}+1}{2-1}-\frac{\sqrt{2}-1}{2-1}$$=\sqrt{2}+1-\sqrt{2}+1$$=2$