Solve $\int (4x+2)\sqrt{x^2+x+1}\,dx$ Trying to solve this for a while now, so far I was able to come up without a proper answer.
Problem : $\displaystyle \int (4x+2)\sqrt{x^2+x+1}\,dx$. 
I tried to take two common from $(4x+2)$ and also to take $(x+1)^2 - x$ from the root, but wasn't able to come up with something to take for substitution. A hint in the right direction would be highly appreciated
Edit :
I forgot to mention this. As this is part of the integration by substitution exercise it'd be highly appreciated if you could provide the hint in that direction.

I'm extremely new to integration and my methods are sloppy but I'd like to try out this question. The following isn't an answer but a demonstration that any substitution can do the job as long as it is done right (Although, a fundamental zen of calculus dictates that the rightest way is the fastest way, I think any path is better than no path)
I'm going to try my luck with $u = 4x+2 \implies \frac{du}{dx} = 4 \implies dx= \frac{1}{4} du$
Also, by the above assumption, $x = \frac{u-2}{4}$ 
$$
\require{cancel}
\begin{align}
&\int (4x+2)\sqrt{x^2+x+1}\,dx \\
&= \int{u\cdot\sqrt{\left(\frac{u-2}{4}\right)^2 + \frac{u-2}{4}+ 1}}\cdot\frac{1}{4} du\\
&= \frac{1}{4} \int{u}\cdot\sqrt{\frac{u^2 - 4u + 4 +4(u-2) + 4^2}{4^2}}\, du\\
&= \frac{1}{4} \int{ u \cdot \frac{1}{4}\cdot \sqrt{u^2 \cancel{- 4u} + 4 \cancel{+ 4u} - 8 +16}}\, du\\
&=\frac{1}{16} \int{u\cdot\sqrt{u^2 + 12}}\, du
\end{align}
$$
Mhh, we need to substitute again inorder to proceed. Let's continue this with
$$v =\sqrt{ u^2 + 12 } 
\implies \frac{dv}{du} = \frac{2u}{2\sqrt{u^2 + 12}} = \frac{u}{v}
\implies du = \frac{v}{u}\, dv$$
Continuing,
$$
\begin{align}
&\frac{1}{16} \int{u\cdot\sqrt{u^2 + 12}}\, du \\
&= \frac{1}{16} \int{u\cdot v \cdot \frac{v}{u}\, dv} \\
&= \frac{1}{16}\int{v^2}\, dv\\
&= \frac{1}{16}\frac{v^{2+1}}{(2+1)} + C\\
&= \frac{v^3}{48} + C
\end{align}$$
Putting $v$ interms of $x$, you'll probably get the right answer.
$\dots$ Maybe taking $u=x^2+x+1$ would have been a better substitution.
Well, practice makes perfect! :D
