Solution verification $\left(A \cap B\right) \cup \left(A \cap B^c\right)$

No, your answer has a mistake

The mistake is in this past

$[(A∩B)∪A]∪[(A∩B)∪B^c]$

The correct one is

$[(A∩B)∪A]∩[(A∩B)∪B^c]$

You can solve the equation easily

$(A∩B)∪(A∩B^c)$

$=(A∪(A∩B^c))∩(B∪(A∩B^c))$

$=A∩(B∪A)$

$=A$

Your first line is incorrect. Remember, in general, $\color{blue}X \color{blue}\cup (Y\cap Z)=(\color{blue}X \color{blue}\cup Y)\cap(\color{blue}X \color{blue}\cup Z)$. You wrote

$$\color{blue}{\left(A \cap B\right)} \color{blue}\cup \left(A \cap B^c\right)=\left[\color{blue}{\left(A \cap B\right)} \color{blue}\cup A\right]\color{red}{\cup }\left[\color{blue}{\left(A \cap B\right)} \color{blue}\cup B^c\right],$$

but the red $\cup$ should be a $\cap$.