A polynomial is an expression which contains a variables, constants and exponents that are combined using the mathematical operators. The given is the addition of two polynomials. The addition can be performed by grouping the like terms. Like terms are terms whose variables and corresponding exponents are same but coefficients need not be same.

Like terms are the terms whose variables and their corresponding exponents are same but coefficients need not be same. The like terms can be added by taking the coefficients and adding them to get the result. The given subtraction expression is simplified by grouping the like terms and then simplified as follows.

\(\displaystyle{\left({14}{x}^{{3}}-{5}{x}^{{2}}+{x}-{9}\right)}-{\left({4}{x}^{{3}}-{3}{x}^{{2}}-{7}{x}+{1}\right)}={14}{x}^{{3}}-{5}{x}^{{2}}+{x}-{9}-{4}{x}^{{3}}+{3}{x}^{{2}}+{7}{x}-{1}\) - Remove the paraenthesis

\(\displaystyle={\left({14}{x}^{{3}}-{4}{x}^{{3}}\right)}+{\left(-{5}{x}^{{2}}+{3}{x}^{{2}}\right)}+{\left({x}+{7}{x}\right)}+{\left(-{9}-{1}\right)}\) Group the like terms.

\(\displaystyle={\left({14}-{4}\right)}{x}^{{3}}+{\left(-{5}+{3}\right)}{x}^{{2}}+{\left({1}+{7}\right)}{x}+{\left(-{9}-{1}\right)}\)

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

Hence, the result of the expression \(\displaystyle{\left({14}{x}^{{3}}-{5}{x}^{{2}}+{x}-{9}\right)}-{\left({4}{x}^{{3}}-{3}{x}^{{2}}-{7}{x}+{1}\right)}\) asked in the question is equal to \(\displaystyle{10}{x}^{{3}}-{2}{x}^{{2}}+{8}{x}-{10}\)