First, find the sum of the polynomials \(\displaystyle-{6}{x}^{{2}}{y}^{{2}}-{x}^{{2}}-{1}\) and \(\displaystyle{5}{x}^{{2}}{y}^{{2}}+{2}{x}^{{2}}-{1}\)

Then,

\(\displaystyle{\left(-{6}{x}^{{2}}{y}^{{2}}-{x}^{{2}}-{1}\right)}+{\left({5}{x}^{{2}}{y}^{{2}}+{2}{x}^{{2}}-{1}\right)}=-{6}{x}^{{2}}{y}^{{2}}+{5}{x}^{{2}}{y}^{{2}}-{x}^{{2}}+{2}{x}^{{2}}-{1}-{1}\)

\(\displaystyle={\left(-{6}+{5}\right)}{x}^{{2}}{y}^{{2}}+{\left(-{1}+{2}\right)}{x}^{{2}}-{2}\)

\(\displaystyle=-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}\)

Therefore, the sum of the polynomials \(\displaystyle-{6}{x}^{{2}}{y}^{{2}}-{x}^{{2}}-{1}\) and \(\displaystyle{5}{x}^{{2}}{y}^{{2}}+{2}{x}^{{2}}-{1}\) is \(\displaystyle-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}\)

Now, subtract \(\displaystyle{9}{x}^{{2}}{y}^{{2}}-{3}{x}^{{2}}-{5}\) from \(\displaystyle-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}\)

So,

\(\displaystyle{\left(-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}\right)}-{\left({9}{x}^{{2}}{y}^{{2}}-{3}{x}^{{2}}-{5}\right)}=-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}-{9}{x}^{{2}}{y}^{{2}}+{3}{x}^{{2}}+{5}\)

\(\displaystyle=-{x}^{{2}}{y}^{{2}}-{9}{x}^{{2}}{y}^{{2}}+{x}^{{2}}+{3}{x}^{{2}}-{2}+{5}\)

\(\displaystyle={\left(-{1}-{9}\right)}{x}^{{2}}{y}^{{2}}+{\left({1}+{3}\right)}{x}^{{2}}+{5}-{2}\)

\(\displaystyle=-{10}{x}^{{2}}{y}^{{2}}+{4}{x}^{{2}}+{3}\)

Hence, the required polynomial is \(\displaystyle-{10}{x}^{{2}}{y}^{{2}}+{4}{x}^{{2}}+{3}\)

Then,

\(\displaystyle{\left(-{6}{x}^{{2}}{y}^{{2}}-{x}^{{2}}-{1}\right)}+{\left({5}{x}^{{2}}{y}^{{2}}+{2}{x}^{{2}}-{1}\right)}=-{6}{x}^{{2}}{y}^{{2}}+{5}{x}^{{2}}{y}^{{2}}-{x}^{{2}}+{2}{x}^{{2}}-{1}-{1}\)

\(\displaystyle={\left(-{6}+{5}\right)}{x}^{{2}}{y}^{{2}}+{\left(-{1}+{2}\right)}{x}^{{2}}-{2}\)

\(\displaystyle=-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}\)

Therefore, the sum of the polynomials \(\displaystyle-{6}{x}^{{2}}{y}^{{2}}-{x}^{{2}}-{1}\) and \(\displaystyle{5}{x}^{{2}}{y}^{{2}}+{2}{x}^{{2}}-{1}\) is \(\displaystyle-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}\)

Now, subtract \(\displaystyle{9}{x}^{{2}}{y}^{{2}}-{3}{x}^{{2}}-{5}\) from \(\displaystyle-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}\)

So,

\(\displaystyle{\left(-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}\right)}-{\left({9}{x}^{{2}}{y}^{{2}}-{3}{x}^{{2}}-{5}\right)}=-{x}^{{2}}{y}^{{2}}+{x}^{{2}}-{2}-{9}{x}^{{2}}{y}^{{2}}+{3}{x}^{{2}}+{5}\)

\(\displaystyle=-{x}^{{2}}{y}^{{2}}-{9}{x}^{{2}}{y}^{{2}}+{x}^{{2}}+{3}{x}^{{2}}-{2}+{5}\)

\(\displaystyle={\left(-{1}-{9}\right)}{x}^{{2}}{y}^{{2}}+{\left({1}+{3}\right)}{x}^{{2}}+{5}-{2}\)

\(\displaystyle=-{10}{x}^{{2}}{y}^{{2}}+{4}{x}^{{2}}+{3}\)

Hence, the required polynomial is \(\displaystyle-{10}{x}^{{2}}{y}^{{2}}+{4}{x}^{{2}}+{3}\)