a. Apply the Product Rule. Let $u=\left(2 x^{2}+3\right)$ and $v=\left(8 x+5+\frac{1}{x}\right)$
\[
\frac{d}{d x}(u v)=\left(2 x^{2}+3\right)(\square)+\left(8 x+5+\frac{1}{x}\right)
\]
Answer
The derivative of the product of \(u\) and \(v\) is \(\boxed{4 x\left(8 x+5+\frac{1}{x}\right)+\left(8-\frac{1}{x^{2}}\right)\left(2 x^{2}+3\right)}\)
Step 1 ：Let \(u=2 x^{2}+3\) and \(v=8 x+5+\frac{1}{x}\)
Step 2 ：Apply the Product Rule of differentiation, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Step 3 ：Find the derivative of \(u\) and \(v\), denoted as \(du\) and \(dv\) respectively. \(du = 4x\) and \(dv = 8 - \frac{1}{x^{2}}\)
Step 4 ：Substitute \(u\), \(v\), \(du\), and \(dv\) into the Product Rule formula: \(du*v + u*dv\)
Step 5 ：The derivative of the product of \(u\) and \(v\) is \(\boxed{4 x\left(8 x+5+\frac{1}{x}\right)+\left(8-\frac{1}{x^{2}}\right)\left(2 x^{2}+3\right)}\)
