Phrasal verbs are a big problem for non-native speakers and that’s completely understandable. There is often no logical explanation for their form. This mistake above though is more of a mix-up between two common phrasal verbs – ‘take care of’ and ‘care about’.
‘Take care of’ means to ‘look after’ (another phrasal verb!) or to be responsible for someone or something. So, when the parents leave and you are left alone with the baby, you need to take care of it. We can also take care of our health (meaning you don’t smoke or drink too much) or take care of a situation (meaning to deal with it/solve it).
‘Care about’ means take an active personal interest in something or someone. So, if you are interested in politics, you can say that you care about politics, or you care about what happens to your country.
To ‘take care about’ does not exist.
Here’s both correct phrasal verbs used together:
A: Hey, you care about what’s happening in Iraq, right? Well, there’s a live discussion at the town hall tomorrow.
B: Oh no, I can’t go! I have to take care of my sister tomorrow night.