<s>[INST] Complex Numbers - Given $(a+b) +i(a-b) = (1+i)^2 + i(2+i)$ obtain the values of $a$ and $b$. This would be a very easy complex number question to someone I understand of most of it its just one of those questions I should know but I've stared at it so much I'm stuck! could someone please explain it to me! here it is:
Question - Given $(a+b) +i(a-b) = (1+i)^2 + i(2+i)$ obtain the values of $a$ and $b$. Note $i=\sqrt{(-1)}$ 
Thank you all! hope to hear from you soon!
 [/INST] Just do it.
$(1+i)^2 + i(2+i) =$
$(1 + 2i + i^2)+(2i + i^2)=$
$(1 + 2i -1) + (2i - 1)=$
$-1 + 4i$
So $-1 + 4i = (a+b) + (a+b)i$
So $a + b = -1$ and $a+b = 4$.
Well, no wonder you are stuck!  You were given an impossible equation.
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From your commments it sounds like what you we actually given was
$(a+b) + (a-b)i = (1+i)^2 + i(2+i)$
or maybe something else?
You still do it the same way:
$(1+i)^2 + i(2+i) = -1 + 4i$
so $(a+b) +(a-b)i = -1 + 4i$
so $a+b = -1$ and $a-b = 4$.
This is solvable. 
$a = - 1 - b$
$-1-b -b = 4$
$-2b = 5$
$b = -5/2$
$a = -1 -(-5/2) = 3/2$.
or....
$a + b = -1$
$a - b = 4$
$(a + b) + (a-b) = -1 + 4$
$2a = 3$
$a = 3/2$
$(a +  b) - (a-b) = -1 -4$
$2b = -5$
$b = -5/2$.
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