Factorising into single and double brackets

Task goals

• Extract a numeric highest common factor into a single bracket, e.g. $5x+15$ or $6x-18$.
• Extract a common factor of $x$ (or a higher power of $x$) into a single bracket, e.g. $x^2+4x$ or $x^2-6x$.
• Extract a combined numeric-and-variable highest common factor, e.g. $10x^2+15x$ or $15x^2+25x$.
• Factorise an expression with a negative leading term, e.g. $-8x-12$ or $-5x+25$.
• State the highest common factor of two given terms, e.g. $12x^2$ and $18x$, and verify a single-bracket factorisation by expanding it back.
• Factorise an all-positive trinomial by finding a pair of numbers that add to $b$ and multiply to $c$, e.g. $x^2+9x+14$.
• Factorise a trinomial where $b$ is negative and $c$ is positive (both factors negative), e.g. $x^2-11x+24$.
• Factorise a trinomial where $c$ is negative (factors have opposite signs), e.g. $x^2+6x-16$ or $x^2-2x-15$.
• Use trinomial factorising in a context problem, e.g. finding the missing side of a rectangle given its area as $x^2+11x+30$ and one side as $(x+5)$.
• Explain why factorising and expanding are inverse processes, using a worked trinomial example.
• Factorise a perfect-square trinomial, e.g. $x^2+14x+49$ or $x^2-8x+16$, and explain why $(x+7)^2$ is not the same as $x^2+49$.
• Factorise a difference-of-squares expression, e.g. $x^2-36$ or $9x^2-25$.
• Fully factorise an expression requiring an HCF extraction before a trinomial or difference-of-squares pattern, e.g. $4x^2+16x-48$ or $3x^2-27$.
• Use fully-factorised forms in a context problem, e.g. finding both side-length expressions of a rectangle given its area as $x^2+2x-15$.