Expanding products and combining like terms

Distribute the second bracket over both terms of the first. To multiply $(a+b)$ and $(c+d)$, treat $(c+d)$ as a single block and distribute it across $a$ and $b$ separately, then distribute again inside each piece:

where: $a$ and $b$ are the two terms of the first bracket, $c$ and $d$ are the two terms of the second bracket, and $ac, ad, bc, bd$ are the four pairwise products that always result from multiplying two binomials.

FOIL is a memory aid, not a new rule. Multiply the First terms, the Outer terms, the Inner terms, and the Last terms, then add all four products. It is just a name for applying $x(y+z) = xy+xz$ twice in a row — the picture below shows why the four products cover the whole rectangle with no gaps and no overlaps.

Like terms have identical letter-power combinations. Two terms can be combined only if they carry exactly the same letters raised to exactly the same powers; their numerical coefficients may differ. For example, $3xy$ and $-7xy$ are like terms, but $3xy$ and $3x^2y$ are not, because the power of $x$ differs. To combine like terms, add or subtract their coefficients and keep the letter part unchanged:

Whether products merge depends on the letters involved. Multiplying two binomials always produces exactly four term-by-term products, because each of the two terms in the first bracket pairs with each of the two terms in the second. Whether those four collapse to three terms, stay as four, or (in special identities met later in the chapter) collapse further depends entirely on whether the Outer and Inner products happen to share the same letters. If the two binomials involve completely different letters, such as $(x+3)(y+5)$, none of the four products can ever combine, and the expanded answer stays a genuine four-term expression.

Expand and simplify $(x+5)(x+3)$.

Step 1. Apply the general formula $ac+ad+bc+bd$ with $a=x$, $b=5$, $c=x$, $d=3$: First $x\times x = x^2$, Outer $x\times 3 = 3x$, Inner $5\times x = 5x$, Last $5\times 3 = 15$.

Step 2. Write all four products together: $x^2 + 3x + 5x + 15$.

Step 3. Identify like terms: $3x$ and $5x$ are like terms, since both carry the letter $x$ raised to power $1$; $x^2$ and $15$ have nothing else in the expression to combine with.

Step 4. Combine the like terms by adding their coefficients: $3x + 5x = (3+5)x = 8x$, giving the final simplified expansion $x^2 + 8x + 15$.

Answer: $(x+5)(x+3) = x^2 + 8x + 15$.

Competitive stretch

Expand and simplify $(2p-3q)(p+4q)$, where the negative sign inside a bracket needs careful sign-tracking.

Step 1. Apply the distributive law twice: First $2p\times p = 2p^2$, Outer $2p\times 4q = 8pq$, Inner $(-3q)\times p = -3pq$, Last $(-3q)\times 4q = -12q^2$.

Step 2. Write all four products: $2p^2 + 8pq - 3pq - 12q^2$.

Step 3. Identify like terms: $8pq$ and $-3pq$ share the letters $p$ and $q$, each to power $1$, so they combine to $8pq - 3pq = 5pq$.

Step 4. Final simplified expression: $2p^2 + 5pq - 12q^2$. Notice this has three terms, even though the expansion started with four products — one Outer-Inner pair merged.

• Skipping one of the four FOIL products, most often the Inner product, leaving a three-term answer that is missing a middle term.
• Combining unlike terms by mistake, such as adding $x^2$ and $8x$ as if they were the same kind of term — only identical letter-and-power combinations may be merged.
• Sign errors during Outer or Inner steps when a bracket contains a subtraction, for instance forgetting that $(-3q)\times p$ gives a negative product.
• Stopping after expansion without combining like terms, leaving an unfinished answer like $x^2+3x+5x+15$ that is correct but not fully simplified.

Ask the tutor: if an expansion is not matching the expected answer, ask the AI tutor to check your FOIL step by step and pinpoint exactly where a sign or a term went missing.