haha ga isogasii toki, watasi ha araimono wo site agemasu.

母（はは） is the plain word for my mother when talking about her to someone else.

A very common pattern in Japanese is:

• 母 / 父 / 兄 / 姉 for your own family when speaking to others
• お母さん / お父さん / お兄さん / お姉さん when referring to someone else’s family, or when directly addressing your own mother/father

So in this sentence, 母 is natural because the speaker is talking about their own mother.

Because 母が忙しい is a clause meaning mother is busy, and that whole clause modifies とき:
母が忙しいとき = when my mother is busy

Here, が marks the subject inside that smaller clause.

Then in the main clause, 私は sets the main topic:
私は洗い物をしてあげます = as for me, I do the washing for her

So the sentence has:

• subordinate clause: 母が忙しいとき
• main clause: 私は洗い物をしてあげます

This is very common in Japanese: が inside the embedded clause, は in the main clause.

Because 忙しい is modifying とき, and とき is a noun meaning time / when.

Japanese i-adjectives can directly modify nouns:

• 忙しい人 = a busy person
• 忙しい日 = a busy day
• 忙しいとき = the time when someone is busy / when someone is busy

So 忙しい stays in its plain form here.

忙しく is the adverbial form, used before verbs, not before nouns.

In Japanese, adjectives like 忙しい can act like predicates by themselves. You do not need a separate word for is in the same way English does.

So:

• 母が忙しい = my mother is busy
• 母が忙しいとき = the time when my mother is busy / when my mother is busy

You can think of とき as attaching to a whole clause.

洗い物（あらいもの） usually refers to things to wash or the washing-up.

In everyday life, it often means:

• dirty dishes
• dishwashing / washing the dishes

Because the sentence says 洗い物をする, the meaning is to do the washing-up, usually to wash the dishes.

So here it does not just mean random washed items—it means the chore.

Because 洗い物 is treated as the object of the verb する.

• 洗い物 = the washing-up / dishes to wash
• をする = to do

So:

• 洗い物をする = to do the washing-up

This is a very common Japanese pattern:

• 勉強をする = do study / study
• 掃除をする = do cleaning / clean
• 洗濯をする = do laundry
• 洗い物をする = do the dishes

〜てあげます means to do something for someone as a favor, kindness, or helpful action.

So:

• 洗い物をします = I do the dishes
• 洗い物をしてあげます = I do the dishes for her

The あげます adds the sense that the action benefits someone else—in this case, the mother.

It is not just stating the action; it shows helping.

Yes, it can.

In this sentence, the speaker is doing something helpful for their mother, so してあげます is natural.

However, learners should know one nuance: 〜てあげる can sometimes sound like the speaker is emphasizing I’m doing this favor for someone, so depending on context, it may sound a little self-conscious if overused.

But in a simple sentence about helping your mother when she is busy, it sounds fine and natural.

Yes, it could definitely be omitted.

Japanese often leaves out subjects and topics when they are clear from context. So this would also be natural:

母が忙しいとき、洗い物をしてあげます。

Adding 私は does one of these things:

• makes the subject clearer
• adds contrast, like as for me
• gives the sentence a more textbook-style completeness

So 私は is not wrong at all, but in real conversation it is often omitted if already understood.

Japanese often puts the time clause or condition clause first, and then gives the main action.

So:

• 母が忙しいとき、私は洗い物をしてあげます。

is very natural:
When my mother is busy, I do the dishes for her.

You could sometimes rearrange things in Japanese, but this order is the standard and easiest structure.

By itself, it often sounds like a general habit or usual behavior:

When my mother is busy, I do the dishes for her.

That is because the sentence uses non-past forms:

• 忙しい
• してあげます

Japanese often relies on context to show whether something is habitual, present, or future.

So without extra context, many learners should understand this as a general statement rather than one specific event.

Because the whole sentence is in polite style.

• してあげます = polite
• してあげる = plain

Japanese usually keeps a consistent style in a sentence. Since this is likely a textbook example or polite speech, ます is used.

If you were speaking casually, you might say:

母が忙しいとき、私は洗い物をしてあげる。

or more naturally in many casual situations:

母が忙しいとき、洗い物してあげる。

Yes, two small things are worth noticing:

1. Family terms change depending on who you are talking about.
   Using 母 for your own mother when speaking to others is very standard Japanese usage.

2. Helping expressions matter.
   〜てあげます does not just describe an action; it frames it as something done for another person’s benefit. Japanese often pays attention to who benefits from an action.

So this sentence is a good example of both family-reference vocabulary and benefactive grammar.