'Shall I carry that bag for you, Pauline?' said John. (offered)
John offered to carry the bad for her.
I believed you were the murderer because of this clue. (led)
This clue led me to believe you were the murderer.
I never imagined the mountains to be/being so high!
to be
Sorry I'm late, I had to stop to pick up/picking up the children from school.
to pick up
Did you manage to find/finding the book you were looking for?
to find
I think that this is the right street. (appears)
This appears to be the right street.
It's very kind of you to give me a lift. (appreciate)
I appreciate you giving me a lift.
Jean succeeded in finishing all her work on time. (managed)
Jean managed to finish all her work in time.
Don't forget to wake me/waking me before you leave.
to wake me
Have you considered to buy/buying a microwave oven?
buying
What do you mean to do/doing about the leaky pipes?
to do
I tried taking/to take that medicine you gave me but I couldn't swallow it.
taking/to take
We have postponed to tell/telling anyone the news until after Christmas,
telling
Margaret was slow at school, but she went on to be/being Prime Minister
to be
I regret to tell you/telling you that we cannot accept your offer.
to tell you