Factorising quadratics with a common factor first

Task goals

• Find the HCF of all three terms of a quadratic, including a negative HCF (or -1) when the leading coefficient is negative, so the quadratic left inside starts positive.
• Write the HCF in front of a bracket containing the smaller quadratic, then factorise that quadratic using the a=1 two-numbers method when it is monic.
• Factorise the quadratic left inside the bracket using an a!=1 method (the pair that multiplies to ac and adds to b) when a!=1 remains after the HCF is removed.
• Write the HCF in front of the finished brackets and check the answer is fully factorised, with no common factor left inside either bracket.
• Explain why an almost-finished factorisation is not fully factorised, find an unknown HCF from a given factorised form, and apply factorising with a common factor to a rectangle-area context.