In the above language w,x belongs to (a+b)+ so w and x can take any value like a,b,ab,ba,aa,bb,aba,bab,bba,bbb,aaa,aab,bba......
Here the main thing to notice is that the language will contain strings which is made up of total three parts in which the first part and the last part are same and the middle part can be anything like {aaa,aba,bab,bbb,bbabb, aabbbaa,abbbab, baabba ....}
which can be seen like this a-a-a, a-b-a, b-a-b, b-bab-b, aa-bbb-aa, ab-bbb-ab, ba-ab-ba where first and third parts are identical so we need some matching mechanism because of strings of type ab-bbb-ab and ba-ab-ba we cant solve it using PDA so we need TM for this so the language is not regular.