Expanding (a+b)(c+d): the 'each with each' method that forgets nothing

The rule

$(a + b)(c + d) = ac + ad + bc + bd$

Every term in the 1st parenthesis multiplies every term in the 2nd.

The “arrows” method (4 steps)

Draw 4 arrows, one per product (aka FOIL):

1. a → c: First × First
2. a → d: Outer (first × last)
3. b → c: Inner (last × first)
4. b → d: Last × Last

Then sum the 4 results.

Example 1: $(x + 3)(x + 5)$

• $x \times x = x^2$
• $x \times 5 = 5x$
• $3 \times x = 3x$
• $3 \times 5 = 15$

Sum: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$

Example 2 (with signs): $(2x - 3)(x - 4)$

• $2x \times x = 2x^2$
• $2x \times (-4) = -8x$
• $(-3) \times x = -3x$
• $(-3) \times (-4) = +12$

Sum: $2x^2 - 8x - 3x + 12 = 2x^2 - 11x + 12$

⚠️ Watch signs: $(-3) \times (-4) = +12$, not $-12$.

Special cases: notable identities

If the two parentheses are identical or opposite, use directly:

• $(a + b)^2 = a^2 + 2ab + b^2$
• $(a - b)^2 = a^2 - 2ab + b^2$
• $(a + b)(a - b) = a^2 - b^2$

Faster than the general method.

Simplify at the end

Always combine like terms: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$.

Common mistake

Forgetting one of the 4 products. Fix: physically draw the 4 arrows on your paper, don’t do it in your head.

?À toi de jouer

Expand (2x + 1)(x − 3).

A.2x² − 5x − 3B.2x² + 5x − 3C.2x² − 3D.2x² − 6x − 3

Valider