You have two questions. The first one is:

MR = d(Total revenue)/dQuantity = d (Price * Quantity)/dQuanitity This is the same as this  : MR = P(Q) + dP/dQ

As Herr K. points out, an important thing to note is that $P$ and $Q$ are not independent of each other; in this context $P$ is a function of $Q$. After this observation, the answer is purely mathematical: the product rule. From Wikipedia:

In calculus, the product rule (or Leibniz rule1 or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as

$ (u\cdot v)' = u'\cdot v+u\cdot v'$

Your second question seems to be

MR = P(Q) * (1+ 1/elasticity(Q))
And then it is converted to:
MR = P(Q) * (1- absolute value of (1/elasticity(Q))
[...] converting the parenthesis in negative I guess its because...
...I would appreciate a deeper explanation...

It is assumed that $P(Q)$ is decreasing in $Q$, and that both $P$ and $Q$ are positive. The (point) elasticity formula is $$ \epsilon_P(Q) = \frac{\text{d}P(Q)}{\text{d}Q} \frac{Q}{P}. $$ The first fraction is negative (because $P(Q)$ is decreasing in $Q$), and the second fraction is positive (both $P$ and $Q$ are positive). Thus the quantity elasticity of price is indeed negative, and hence $$ \epsilon_P(Q) = - | \epsilon_P(Q) |. $$

You have two questions. The first one is (I broke lines in the quotes where I tought this improved legibility):

MR = d(Total revenue)/dQuantity = d (Price * Quantity)/dQuanitity 
This is the same as this:
MR = P(Q) + dP/dQ

As Herr K. points out, an important thing to note is that $P$ and $Q$ are not independent of each other; in this context $P$ is a function of $Q$. After this observation, the answer is purely mathematical: the product rule. From Wikipedia:

In calculus, the product rule (or Leibniz rule1 or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as

$ (u\cdot v)' = u'\cdot v+u\cdot v'$

Your second question seems to be

MR = P(Q) * (1+ 1/elasticity(Q))
And then it is converted to:
MR = P(Q) * (1- absolute value of (1/elasticity(Q))
[...] converting the parenthesis in negative I guess its because...
...I would appreciate a deeper explanation...

It is assumed that $P(Q)$ is decreasing in $Q$, and that both $P$ and $Q$ are positive. The (point) elasticity formula is $$ \epsilon_P(Q) = \frac{\text{d}P(Q)}{\text{d}Q} \frac{Q}{P}. $$ The first fraction is negative (because $P(Q)$ is decreasing in $Q$), and the second fraction is positive (both $P$ and $Q$ are positive). Thus the quantity elasticity of price is indeed negative, and hence $$ \epsilon_P(Q) = - | \epsilon_P(Q) |. $$