Eradicate native Australian solicitor (4,3)

I believe the answer is:

root out

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'eradicate' is the definition.
(I know that eradicate can be written as root out)

'native australian solicitor' is the wordplay.
'native australian' becomes 'roo' (I can't explain this - if you can you should give a lot more credence to this answer).
'solicitor' becomes 'tout' (someone who solicits business).
'roo'+'tout'='ROOT OUT'

Can you help me to learn more?

(Other definitions for root out that I've seen before include "Extirpate, find and destroy" , "Destroy completely" , "Completely remove" , "Extirpate, eradicate" , "Get rid or, extirpate" .)

I've seen this clue in The Telegraph.

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