a lot of or lots of

a lot of and lots of are two very useful, common phrases.
(We never spell them a lots of, alot of.) 

Answers to the exercise above: c. and d. are correct.

a lot of (3 separate words) and lots of (two separate words) mean the same thing. We use them before nouns to mean ‘a large number or amount of'. 

a lot of is slightly more formal than lots of, but we don't use them in academic and business writing - see note from Oxford Dictionary below*

Remember
a lot of = three individual words 
lots of = two individual words

We never write 'alot of' or 'a lots of'.

I've got a lot of shopping.
I've got lots of shopping.
There are a lot of people.
There are lots of people. 
He always eats a lot of food.
He always eats lots of food.

A lot of is three individual words.

Pronunciation can confuse people with the spelling of it.

1. We have the stressed 'a lot of'
2. or unstressed "a lottuv" - the 't' joins the 'o' to make "tuv" "a lottuv"
3. or even more unstressed "a lotta"

1. We had a lot of fun.
2. We had "a lottuv" fun. (used in speaking only)
3. We had "a lotta" fun. (speaking, and used in some very informal writing)

1. It's a lot of money.
2. It's "a lottuv" money.
3. It's "a lotta" money. 

Or even more unstressed in some dialects like Liverpool "a lora" money.

It doesn't matter how you pronounce it, all dialects and how we say a lot of is valid, but there's only one way to spell it - a lot of.

lots of is two words and is more informal, conversational.

The pronunciation is like "a lot of" with various unstressed forms.
1. We have the stressed "lots of"
2. Unstressed - the 's' joins the 'o' "lotsuv"
3. or even more unstressed "lotsa". The Oxford Dictionary has this very informal version on its website - "He's got lotsa money."

*A quick note about a lot of and lots of from the Oxford Dictionary: A lot of and lots of are very common in speech and writing but they still have an informal feel and are generally not considered acceptable for formal, business, academic English, where alternatives such as many or a large number, a great deal of are used instead.

So remember it doesn't matter how you say these but make sure you write them as a lot of or lots of.

We can also have a lot and lots.

Thanks a lot or Thanks very much or Thanks so much

= a large number or amount

Check out these sentences and definitions from Oxford Learner Dictionary:

Thanks a lot.

I miss my family a lot.

• ‘How many do you need?’ ‘A lot.’

• Have some more cake. There's lots left.

• He has invited nearly a hundred people but a lot aren't able to come.

• a lot to do I have a lot to do today.