We need to find the given sum of polynomials

\(\displaystyle{\left(-{10}{z}^{{4}}+{8}{x}^{{3}}-{x}^{{2}}+{10}{x}\right)}+{\left(-{8}{x}^{{3}}-{5}{x}^{{2}}-{4}\right)}\)

Remove the parantheses using (-a)=-a

\(\displaystyle-{10}{x}^{{4}}+{8}{x}^{{3}}-{x}^{{2}}+{10}{x}-{8}{x}^{{3}}-{5}{x}^{{2}}-{4}\)

Now group the like terms (terms with same exponent of x)

\(\displaystyle-{10}{x}^{{4}}+{8}{x}^{{3}}-{8}{x}^{{3}}-{x}^{{2}}-{5}{x}^{{2}}+{10}{x}-{4}\)

Now add the like terms

\(\displaystyle-{10}{x}^{{4}}+{\left({8}{x}^{{3}}-{8}{x}^{{3}}\right)}+{\left(-{x}^{{2}}-{5}{x}^{{2}}\right)}+{10}{x}-{4}\)

\(\displaystyle={\left(-{10}{x}^{{4}}-{6}{x}^{{2}}+{10}{x}-{4}\right)}\)

The sum of the polynomials is \(\displaystyle{\left(-{10}{x}^{{4}}-{6}{x}^{{2}}+{10}{x}-{4}\right)}\)

\(\displaystyle{\left(-{10}{z}^{{4}}+{8}{x}^{{3}}-{x}^{{2}}+{10}{x}\right)}+{\left(-{8}{x}^{{3}}-{5}{x}^{{2}}-{4}\right)}\)

Remove the parantheses using (-a)=-a

\(\displaystyle-{10}{x}^{{4}}+{8}{x}^{{3}}-{x}^{{2}}+{10}{x}-{8}{x}^{{3}}-{5}{x}^{{2}}-{4}\)

Now group the like terms (terms with same exponent of x)

\(\displaystyle-{10}{x}^{{4}}+{8}{x}^{{3}}-{8}{x}^{{3}}-{x}^{{2}}-{5}{x}^{{2}}+{10}{x}-{4}\)

Now add the like terms

\(\displaystyle-{10}{x}^{{4}}+{\left({8}{x}^{{3}}-{8}{x}^{{3}}\right)}+{\left(-{x}^{{2}}-{5}{x}^{{2}}\right)}+{10}{x}-{4}\)

\(\displaystyle={\left(-{10}{x}^{{4}}-{6}{x}^{{2}}+{10}{x}-{4}\right)}\)

The sum of the polynomials is \(\displaystyle{\left(-{10}{x}^{{4}}-{6}{x}^{{2}}+{10}{x}-{4}\right)}\)