Hi there. I need to know where is my mistake so I can correct it. Hope you can help me.
Hi
The problem is that $\displaystyle a^2+5a+1^2$ is not a perfect square. A perfect square is something like $\displaystyle x^2+2xy+y^2=(x+y)^2$. If you choose $\displaystyle x=a$ and $\displaystyle y=1$, the term $\displaystyle 2xy$ equals $\displaystyle 2a$, not $\displaystyle 5a$... it doesn't work.
Unfortunately you made a mistake when factorizing
$\displaystyle a^2+5a+1 = \underbrace{a^2+5a+{\bold{\color{red}\dfrac{25}4 - \dfrac{25}4}} + 1}_{completing\ the\ square} = \underbrace{\left(a+\dfrac52\right)^2-\dfrac{21}4}_{sum\ of\ squares} = $$\displaystyle \underbrace{\left[\left(a+\dfrac52\right)+\sqrt{\dfrac{21}4}\right] \left[\left(a+\dfrac52\right)-\sqrt{\dfrac{21}4}\right]}_{factorized} $
And now go ahead!