Step 1
\(\displaystyle{f{{\left({x}\right)}}}={\left({4}{x}+{3}\right)}{\left({x}+{2}\right)}^{{{2}}}\)
The function has two zeros \(\displaystyle-{\frac{{{3}}}{{{4}}}},-{2}\)
So, the graph of f(x) crosses the x-axis at \(\displaystyle{\left({\frac{{{4}}}{{{3}}}},{0}\right)}\) and \(\displaystyle{\left(-{2},{0}\right)}\)
To find the y-intercept, substitute 0 for x in f(x)
\(\displaystyle{f{{\left({x}\right)}}}={\left({4}{x}+{3}\right)}{\left({x}+{2}\right)}^{{{2}}}\)

\(\displaystyle{f{{\left({0}\right)}}}={\left({4}{\left({0}\right)}+{3}\right)}{\left({0}+{2}\right)}^{{{2}}}\) Substitute 0 for x Simplify \(\displaystyle={\left({3}\right)}{\left({2}\right)}^{{{2}}}\)

\(\displaystyle={12}\) So, the function f(x) crosses the y-axis at (0,12) Step 2 \(\displaystyle{4}{x}\cdot{\left({x}\right)}^{{{2}}}={4}{x}^{{{3}}}\) The leading coefficient is 4 Since the leading coefficient is positive and the function f(x) of degree 3 (odd degree) So, the end behavior is \(\displaystyle{x}\rightarrow\infty,{f{{\left({x}\right)}}}\rightarrow\infty\)

\(\displaystyle{x}\rightarrow-\infty,{f{{\left({x}\right)}}}\rightarrow-\infty\) See the graph below

\(\displaystyle{f{{\left({0}\right)}}}={\left({4}{\left({0}\right)}+{3}\right)}{\left({0}+{2}\right)}^{{{2}}}\) Substitute 0 for x Simplify \(\displaystyle={\left({3}\right)}{\left({2}\right)}^{{{2}}}\)

\(\displaystyle={12}\) So, the function f(x) crosses the y-axis at (0,12) Step 2 \(\displaystyle{4}{x}\cdot{\left({x}\right)}^{{{2}}}={4}{x}^{{{3}}}\) The leading coefficient is 4 Since the leading coefficient is positive and the function f(x) of degree 3 (odd degree) So, the end behavior is \(\displaystyle{x}\rightarrow\infty,{f{{\left({x}\right)}}}\rightarrow\infty\)

\(\displaystyle{x}\rightarrow-\infty,{f{{\left({x}\right)}}}\rightarrow-\infty\) See the graph below