A bracket is a parcel you unwrap.

What does 3(x + 2) even mean?

In algebra, when a number sits right up against a bracket with no sign between them, it means multiply. There's an invisible "times" hiding in the gap. So 3(x + 2) reads as "three times the whole thing in the bracket" — three lots of (x + 2).

The bracket is doing an important job: it ties x and 2 together into one bundle, so the 3 has to multiply the bundle as a unit, not just the first thing it bumps into. Picture three identical gift boxes, and inside each one is an x and a 2. Tip them all out and you've got three x's and three 2's: that's x + 2 + x + 2 + x + 2, which tidies up to 3x + 6. Same answer — expanding is just the fast way to do that.

Two words worth pinning down, because your teacher will use them constantly. The thing on the outside is the factor (here, 3). Each separate piece inside the bracket — the x and the 2 — is a term. Expanding means rewriting the expression without brackets by multiplying the factor through every term. We sometimes call it "multiplying out," and it's the exact reverse of factorising (squeezing a common factor back outside a bracket).

Expanding never changes how much you've got. It's the same quantity in different clothes — the parcel and its unwrapped contents are worth exactly the same.

See it as area.

Here's the picture that makes it obvious. Draw a rectangle. Its height is the factor (the number outside). Its length is the bracket, split into its terms. The area of each piece is one term of your answer — and the whole rectangle is the whole answer. Drag the sliders and watch the parcel rebuild itself.

Every term. No exceptions.

The single most important habit in expanding is this: the factor multiplies each and every term inside the bracket, not just the first one it reaches. The area model shows you why with no arguing — both blocks are real rectangles with real area, so both have to be counted.

It helps to imagine little arrows, one shooting from the factor to each term inside. For 3(x + 2) you draw two arrows: 3 → x gives 3x, and 3 → 2 gives 6. Two terms inside means two arrows means two pieces in the answer. If the bracket had three terms, like 2(x + y + 4), you'd fire three arrows: 2x + 2y + 8. The factor visits everyone at the party.

When the parcel has a minus in it.

Real brackets aren't always full of friendly plusses. Sometimes there's a subtraction inside, sometimes a minus sitting outside. The rule never changes — multiply every term — but you have to carry the sign along for the ride. Think of the sign as glued to the term in front of it; wherever the term goes, its sign goes too.

A minus inside the bracket

Take 2(x − 3). The second term isn't 3, it's minus 3. So the factor 2 multiplies x to give 2x, and multiplies −3 to give −6. The answer is 2x − 6. The minus survives the unwrapping.

A minus outside the bracket

Now the sneaky one: a minus sign in front, like −(x + 4). That little dash is really a hidden −1. So you're multiplying everything inside by −1, which simply flips the sign of each term: −x − 4. A bare minus outside a bracket means "swap every sign inside."

And both together, like −2(x − 5): the factor is −2, so −2 × x = −2x and −2 × (−5) = +10 (a minus times a minus is a plus). That gives −2x + 10. Go slowly, keep each sign attached to its term, and signs stop being scary.

Expand first, then collect like terms.

Often a bracket is part of a longer expression, and once you've unwrapped it there's some tidying to do. Like terms are terms of the same type — all the x's together, all the plain numbers together. You can add or subtract like terms, because they're the same kind of thing. (You can't merge an x with a number, just as you can't add 3 apples to 2 oranges and call it 5 of anything.)

So with something like 2(x + 3) + 4x, you expand the bracket first, then sweep up the matching pieces:

The 6 stayed put because there was no other plain number to join it. The two x-terms found each other and combined. Expanding and collecting are a tag team: unwrap every parcel, then gather the same kinds of things into one neat pile.

Does the unwrapped form really equal the parcel?

Here's how you can be certain you expanded correctly: pick any number for x and work out both forms. If your expanding was right, the packed-up 3(x + 2) and the unwrapped 3x + 6 must land on the very same number — every single time. Slide x around and watch them stay locked together.

This is a brilliant safety net. If you ever expand a bracket and you're not sure, plug in an easy number like x = 2 to both the original and your answer. Get two different numbers and you know a mistake crept in — usually a forgotten term or a flipped sign.

What happens with two brackets?

Once single brackets feel easy, here's a glimpse of where this goes next — no need to master it yet, just enjoy that it's the same idea, bigger parcel. Sometimes you multiply two brackets together, like (x + 2)(x + 3). The area model still works perfectly: now both the height and the length are split, so the rectangle breaks into four pieces instead of two.

Add the four areas: x² + 3x + 2x + 6. Then collect the like terms (3x + 2x = 5x) and you get x² + 5x + 6. Exactly the same move you already know — area split into pieces, every piece multiplied, like terms gathered — just with one more split. The parcel got bigger; the unwrapping didn't change.

"3(x + 2) = 3x + 2"? Afraid not.

This is the mistake almost everyone makes at least once: multiply the first term and forget the second one, writing 3(x + 2) = 3x + 2. But look back at the area model — that leaves the entire right-hand block uncounted. The 2 inside also gets multiplied by 3, so it becomes 6, not 2. The honest answer is 3x + 6. Two terms in, two terms out. Let's lock it in.