Answer
\[ = 2{e^{4x}}\,\left( {2{x^2} + 13x + 21} \right)\]
Work Step by Step
\[\begin{gathered}
y = \,{\left( {x + 3} \right)^2}{e^{4x}} \hfill \\
Expand\,\,\,\,{\left( {x + 3} \right)^2} \hfill \\
y = \,\left( {{x^2} + 6x + 9\,} \right)\,{e^{4x}} \hfill \\
Find\,\,the\,\,derivative \hfill \\
{y^,} = \,\,\left[ {\,\left( {{x^2} + 6x + 9} \right){e^{4x}}} \right]{\,^,} \hfill \\
Use\,\,the\,\,product\,\,rule\, \hfill \\
{y^,} = \left( {{x^2} + 6x + 9} \right)\,{\left( {{e^{4x}}} \right)^,} + {e^{4x}}{\left( {{x^2} + 6x + 9} \right)^,} \hfill \\
Use\,\,\,\,\frac{d}{{dx}}\,\,\left[ {{e^{g\,\left( x \right)}}} \right] = {e^{g\,\left( x \right)}}{g^,}\,\left( x \right) \hfill \\
Then \hfill \\
{y^,} = \left( {{x^2} + 6x + 9} \right)\,\left( {4{e^{4x}}} \right) + {e^{4x}}\,\left( {2x + 6} \right) \hfill \\
Simplify\,\,by\,multiplying\,\,and\,\,combining\,\,terms \hfill \\
{y^,} = 4{x^2}{e^{4x}} + 24x{e^{4x}} + 36{e^{4x}} + 2x{e^{4x}} + 6{e^{4x}} \hfill \\
{y^,} = 4{x^2}{e^{4x}} + 26x{e^{4x}} + 42{e^{4x}} \hfill \\
Factor \hfill \\
= 2{e^{4x}}\,\left( {2{x^2} + 13x + 21} \right) \hfill \\
\end{gathered} \]
