Solve the following quadratic equations by factorization:

We have been given

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

Therefore,

$x^{2}+a x+\frac{1}{a} x+1=0$

$x(x+a)+\frac{1}{a}(x+a)=0$

$\left(x+\frac{1}{a}\right)(x+a)=0$

Therefore,

$x+\frac{1}{a}=0$

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

or,

$x+a=0$

$x=-a$

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