Calculation of pH in acid base reaction
No, you should not do the problem your way.
Given
$$(0.08−x)(0.09−x)=10^{−14}$$
You can solve for $x$ here, but its value will be quite close to $0.08$. This is the only way to make the product a small number, by making the one of the multiplicands very small.
However, $x$ will be so close to 0.08 that will have trouble figuring out the value of $0.08 - x$, which is what you want. Therefore, the right answer is obscured by your method of solving the problem.
In this problem, you can work around that by calculating the pH value from the hydroxide ion concentration, which, by the argument above, will be roughly $0.01\ \mathrm{M}$, consistent with a pH of 12. However, in the most general case, you may not be able to do that to determine what the final equilibrium concentrations are.