[tex]\displaystle \\
a) \\
a^2+4a+b^2-6b+13=0 \\
a^2+4a+b^2-6b+4+9=0 \\
a^2+4a+4 + b^2-6b+9=0 \\
(a^2+2\times a\times b +2^2) + (b^2+2\times b \times (-3)+(-3)^2)=0 \\
(a+2)^2 + (b -3)^2 = 0 \\
\text{o expresie ridicata la puterea a 2-a} nu poate fi negativa.}\\
\text{O suma de expresii la puterea a doua } = 0 \text{ daca fiecare expresie }=0 \\ \\
\Longrightarrow ~~ a+2 = 0~~~si~~~b-3=0 \\
a+2 = 0~~ \Longrightarrow ~~ \boxed{a=-2} \\
b-3 = 0~~ \Longrightarrow ~~ \boxed{b=3} \\ [/tex]
[tex]b)\\a^2+b^2+2(a+2b)+5=0\\
a^2+b^2+2a+4b+5=0\\
a^2+2a+b^2+4b+5=0\\
a^2+2a+b^2+4b+1 + 4=0 \\
a^2+2a+1+b^2+4b + 4=0 \\
(a^2+2\times a\times 1+1^2)+(b^2+2\times b \times 2 + 2^2)=0 \\
(a+1)^2+(b+2)^2=0 \\
\text{o expresie ridicata la puterea a 2-a} nu poate fi negativa.} \\
\text{O suma de expresii la puterea a doua } = 0 \text{ daca fiecare expresie }=0 \\ \\
\Longrightarrow ~~a+1=0~~~si~~~b+2=0 \\
a+1 = 0~~ \Longrightarrow ~~ \boxed{a=-1} \\
\\ b+2 = 0~~ \Longrightarrow ~~ \boxed{b=-2} [/tex]
