Solve the following quadratic equations by factorization:

We have been given

$a\left(x^{2}+1\right)-x\left(a^{2}+1\right)=0$

$a x^{2}-\left(a^{2}+1\right) x+a=0$

$a x^{2}-a^{2} x-x+a=0$

$a x(x-a)-1(x-a)=0$

$(a x-1)(x-a)=0$

Therefore,

$a x-1=0$

$a x=1$

$x=\frac{1}{a}$

or,

$x-a=0$

$x=a$

Hence, $x=\frac{1}{a}$ or $x=a$.