Factorising special double brackets (perfect squares, difference between squares)

Task goals

• Factorise a perfect-square trinomial with a leading coefficient of 1, such as $x^2+6x+9$, into $(x+a)^2$ form.
• Factorise a difference of two squares with a leading coefficient of 1, such as $x^2-16$, into $(x-a)(x+a)$ form.
• Fill in a missing number in a partially-given perfect-square or difference-of-squares factorisation, such as $x^2+12x+36=(x+\square)^2$ or $p^2-64=(p-\square)(p+\square)$.
• Factorise a perfect-square trinomial or difference of two squares where the leading coefficient is greater than 1, such as $4y^2+12y+9$ or $25p^2-1$.
• Decide whether a given factorisation is correct, or identify which of four expressions does not belong with the others.
• Factorise fully an expression requiring two applications of difference of squares, such as $x^4-81$.
• Explain how to tell whether a quadratic is a perfect-square trinomial or a difference of two squares.
• Given the area of a square as a quadratic expression, find an expression for its side length.