Factorising double brackets (with a co-efficient equal to 1)

Task goals

• Factorise a quadratic $x^2+bx+c$ into two brackets when both numbers found are positive.
• Factorise a quadratic $x^2-bx+c$ into two brackets when both numbers found are negative.
• Fill in one or both missing numbers in a partially-given factorisation, such as $x^2+8x+15=(x+\square)(x+\square)$.
• Factorise a quadratic where the constant term is negative, so the two brackets have one positive and one negative number, such as $x^2+a-12$ or $x^2-3x-10$.
• Decide whether a given factorisation is correct, or identify which of four expressions does not belong with the others.
• Explain why the two numbers used in the brackets must add to $b$ and multiply to $c$ in $x^2+bx+c$.
• Given the area of a rectangle as a quadratic expression and one side as a binomial factor, find the expression for the other side.