Question: Expand and simplify $(3x - 8y)(2x + 3y) - 3xy$.

Step 1. Expand the double bracket — each term in the first bracket multiplies each term in the second (four products):

Step 2. Write the expansion, keeping the trailing term in sight:

Step 3. Group the like ($xy$) terms:

Step 4. Final simplified form:

Check. Substitute $x = 1, y = 1$: original $(3-8)(2+3) - 3 = -25 - 3 = -28$; expanded $6 - 10 - 24 = -28$. ✓ — a ten-second full-expansion check.

A tidy habit: write all four raw products on one line before collecting — markers credit the expansion even when a collection slip follows.

⚠ Watch out: The sign rides with the term: $-8y \times 2x = -16xy$ and $-8y \times 3y = -24y^2$. The dangling $-3xy$ is deliberately placed to be forgotten — bring every stray term through each line until it merges.

Question: Expand (a) $(2p + 3q)^2$ and (b) $(5m - 2n)^2$ using the algebraic identities.

Step 1. The three identities to know cold:

Step 2. (a) With $a = 2p$, $b = 3q$:

Step 3. (b) With $a = 5m$, $b = 2n$ and the minus version:

Step 4. The third identity turns near-identical brackets into a two-term answer instantly:

— no middle term at all (the difference of two squares).

These identities also run backwards in factorisation and power algebra questions — recognising m^2 - 4n^2 20 20 12 61 79 80 81 98 33 100 204 250 395 398 399 400 7015m)^2 - (2n)^2 the same fact read right-to-left.

⚠ Watch out: $(2p+3q)^2 \ne 4p^2 + 9q^2$ — squaring a bracket is not squaring each term; the middle term $2ab$ is the whole point. Square the coefficient along with the letter ($(2p)^2 = 4p^2$), and note only the $2ab$ term carries the sign.