I watched the news on television about my former (old?) next door neighbor [FORMER IS THE RIGHT WORD BUT YOU COULD ALSO USE "PREVIOUS" BECAUSE "OLD" COULD RELATE TO HIS AGE, SO "OLD" IS NOT THE BEST CHOICE] WHO was arrested this month. Now, two cops who come from the A neighboring city [NOT "THE" BECAUSE THE READER DOES NOT KNOW WHICH CITY IS "THE" NEIGHBORING CITY, SO WE USE "A" TO MEAN 'ANY NEIGHBORING CITY BUT WE DON'T KNOW WHICH CITY] have been investigating(?) [YES, INVESTIGATING IS EXACTLY THE CORRECT VERB TO USE AND YOU COULD SAY "HAVE BEEN INVESTIGATING OR ARE INVESTIGATING] about [DELETE "ABOUT" BECAUSE THE VERB INVESTIGATE IS FOLLOWED BY SOMEONE OR SOMETHING, FOR EXAMPLE: THE BOSS / THE COMPANY] my new next door neighbor for A different case.