Help with Maths (quadratics).?

a²x² – 2ax – a + 1 = a²x² – 2ax + 1 – a
.......................... = (a²x² – 2ax + 1) – (√a)²
.......................... = (ax – 1)² – (√a)²
.......................... = [(ax – 1) + (√a)][(ax – 1) – (√a)]
.......................... = (ax – 1 + √a)(ax – 1 – √a)
.......................... = (ax + √a – 1)(ax – √a – 1)

Start with the right hand side.

ax +√a-1)(ax-√a-1)
Use the distributive law
=ax(ax-√a-1)+√a(ax-√a-1)-1(ax-√a-1)
Multiply out the brackets
=[a^2x^2 - ax√a - ax]+[ax√a -a-√a]-[ax-√a-1]
Collect like terms
=a^2x^2 - ax√a+ax√a-ax-ax -a +1
= a^2x^2 - 2ax - a + 1

a^2x^2 - 2ax - a + 1
= a^2x^2 - 2ax + 1 - a
= a^2x^2 - ax - ax + 1 - a
= (a^2x^2 - ax) - (ax - 1) - a
= ax(ax - 1) - 1(ax - 1) - a
= (ax - 1)(ax - 1) - a
= (ax - 1)^2 - (√a)^2
= [(ax - 1) + √a][(ax - 1) - √a]
= [ax - 1 + √a][ax - 1 - √a]
= [ax + √a - 1][ax - √a - 1]

(m^2-n^2)=(m+n)(m-n)

a^2x^2 - 2ax - a+1 = a^2x^2 - 2ax - (a - 1)

= a^2x^2 - 2ax - (sqrt(a)^2 - 1)

= ( ax + (sqrt(x) - 1)) (ax - (sqrt(x) + 1))

= (ax + squareroot a - 1)(ax - squareroot a -1)

You can show this very easily using the cross method to factorize quadratic equations.