can someone check this for me?
No, both are wrong (first one in the answer, second one in the writing)
$\displaystyle \Rightarrow$ In the first one, you must write both answers :
$\displaystyle x = \frac{4}{\sqrt{5}}$ and $\displaystyle x = - \frac{4}{\sqrt{5}}$
$\displaystyle \Rightarrow$ The second one has the correct answer but is not correctly written. Here is an example of how it should be written :
$\displaystyle x^2 - 8x + 5 = 0$
$\displaystyle x^2 - 8x = -5$
$\displaystyle x^2 - 8x + 16 = 11$
$\displaystyle (x - 4)^2 = 11$
$\displaystyle x - 4 = \sqrt{11}$ or $\displaystyle x - 4 = - \sqrt{11}$
$\displaystyle x = \sqrt{11} + 4$ or $\displaystyle x = - \sqrt{11} + 4$
Thus $\displaystyle x = 4 \pm \sqrt{11}$
EDIT : talking about the first question in this post ...
The other question you posted is wrong too
I am not taking account of the quadratic equation written on it because I do not see the link with the following.
Here is how you do it :
$\displaystyle \frac{-4 \pm \sqrt{40}}{2} = \frac{-4 \pm \sqrt{4 \times 10}}{2}$
$\displaystyle = \frac{-4 \pm 2 \sqrt{10}}{2} = \frac{2(-2 \pm \sqrt{10})}{2}$ --> This is where you messed up : you did not factorize the $\displaystyle -4$.
$\displaystyle = -2 \pm \sqrt{10}$
Or, more conveniently :
$\displaystyle = 2 \mp \sqrt{10}$
Remember that $\displaystyle ab + ac = a(b + c)$. Here, you have :
$\displaystyle \frac{-4 \pm 2 \sqrt{10}}{2}$
This can be written :
$\displaystyle \frac{-2 \times 2 \pm 2 \sqrt{10}}{2}$
The common factor is two, so you factorize it over the addition. Since you are factorizing all the $\displaystyle 2$ (and not $\displaystyle -2$ !), they become $\displaystyle 1$ (some people prefer to remove them directly though) :
$\displaystyle \frac{2(-2 \times 1 \pm 1 \sqrt{10})}{2}$
Now you can cancel $\displaystyle 2$ out :
$\displaystyle -2 \times 1 \pm 1 \sqrt{10}$
Which is simplified to :
$\displaystyle -2 \pm \sqrt{10}$
Or, even more conveniently :
$\displaystyle 2 \mp \sqrt{10}$
alright let's break it down.
We have
$\displaystyle -4\pm2\sqrt10 $
if we had $\displaystyle 2x^2+4x$
we would factorise it by taking 2x out of the equation.
So we divide all terms by 2x:
$\displaystyle x+2$
So same as $\displaystyle -4\pm2\sqrt10 $
we can take 2 out of the equation;
So divide everything by 2
you will get: $\displaystyle -2\pm\sqrt10 $
I divided the -4 by 2 (which made -2) and 2 by 2 ( which made 1)
Is this more clear now?