BAD : It has taken them almost two months to answer to my letter.
GOOD : It has taken them almost two months to answer my letter.
BAD : I couldn't answer to the last two questions.
GOOD : I couldn't answer the last two questions.
BAD : My job involves cooking, cleaning and answering to the door bell.
GOOD : My job involves cooking, cleaning and answering the door bell.
answer sb/sth (WITHOUT to ): 'Have you answered their letter yet?'
Note that answer also has less common meanings where to is necessary: 'Anyone who upsets you will have me to answer to.' 'The dog answers to the name of Zak.'
BAD : I rang the number you gave me but nobody answered me.
GOOD : I rang the number you gave me but nobody answered.
BAD : I called her name but nobody answered me.
GOOD : I called her name but nobody answered.
Do not use an object with answer unless this adds new meaning to the sentence (meaning which is not obvious from the rest of the sentence). Compare: 'He asked her for an explanation but she didn't answer.' 'I'd like to speak to him but he refuses to answer the phone.' In the first sentence the object 'him' is understood and there is no need to mention it.
BAD : They still can't find an answer for this problem.
GOOD : They still can't find an answer to this problem.
an answer to a problem or question (NOT for ): 'The answer to the second question is Abraham Lincoln.'