Trinomial with leading coefficient other than 1

When the coefficient of the second-degree term is other than $1$, we look for two numbers whose sum equals the central coefficient and whose product equals the product of the first by the last coefficient.

Example — Factorise $3x^2 + 2x - 8$

We look for two numbers with sum $= 2$ and product $= 3\cdot(-8) = -24$.

Systematic search: we factorise $24 = 2^3\cdot 3$ and try the pairs of factors.

Pair	Sum	✓
$1,\;24$	$25$
$2,\;12$	$14$
$3,\;8$	$11$
$4,\;6$	$10$
$6,\;-4$	$2$	✓

Splitting the central term and grouping in pairs: $3x^2 + 6x - 4x - 8 = 3x(x+2) - 4(x+2) = \boxed{(x+2)(3x-4)}$

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Topics: Algebraic calculation
Concepts: Partial grouping · Factorisation · Special trinomial
Skills: Factorising