Linearization through tangent line approximation is achieved with equation:

\(\displaystyle{L}{\left({x}\right)}\approx{f{{\left({a}\right)}}}+{f}`{\left({a}\right)}\cdot{\left({x}-{a}\right)}\)

The derivative of f(x) is:

\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{4}}}+{3}{x}^{{{2}}}\)

\(\displaystyle{f}`{\left({x}\right)}={\left({x}^{{{4}}}+{3}{x}^{{{2}}}\right)}`\)

\(\displaystyle={\left({x}^{{{4}}}\right)}`+{\left({3}{x}^{{{2}}}\right)}`\)

\(\displaystyle={4}{x}^{{{3}}}+{2}\cdot{3}{x}^{{{1}}}\)

\(\displaystyle={4}{x}^{{{3}}}+{6}{x}\)

And with \(\displaystyle{f{{\left({a}=-{1}\right)}}}={4}\) and \(f`(a=-1)=-10\) our linearization model is:

\(\displaystyle{L}{\left({x}\right)}={4}-{10}\cdot{\left({x}+{1}\right)}\)

\(\displaystyle={4}-{10}\cdot{x}-{10}\)

\(\displaystyle{L}{\left({x}\right)}=-{6}-{10}\cdot{x}\)