Question

How do you factor `(a^2 +1)^2 - 7(a^2 +1) +10`?

Answer 1

`a_1=1 , a_2=-1, a_3=2, a_4=-2`

Explanation:

Let `(a^2+1) =x`
so the eqn is `x^2-7x+10=0`
now,
`x^2 -5x-2x+10=0`
`=>x(x-5)-2(x-5)=0`
`=>(x-2)*(x-5)=0`
`=>x-2=0 =>x=2 =>a^2+1=2 => a^2=1`
`=>a=+-1`
Again,
`=>x-5=0 =>x=5 =>a^2+1=5 => a^2=4`
`=>a=+-2`

Answer 2

`(a^2+1)^2-7(a^2+1)+10 = (a+1)(a-1)(a+2)(a-2)`

Explanation:

Let `u = a^2+1`, then the expression is:

`u^2-7u+10` which can be factored:

`(u-2)(u-5)`

Replacing `u`, we get:

`((a^2+1)-2)((a^2+1)-5)`.

We can simplify to get:

`(a^2-1)(a^2-4)`.

Each of these is a difference of squares, so we can finish with:

`(a+1)(a-1)(a+2)(a-2)`