Factorise:

$x^{2}+\frac{12}{35} x+\frac{1}{35}=\frac{35 x^{2}+12 x+1}{35}$

$=\frac{35 x^{2}+7 x+5 x+1}{35}$

$=\frac{7 x(5 x+1)+1(5 x+1)}{35}$

$=\frac{(5 x+1)(7 x+1)}{35}$

$=\frac{(5 x+1)(7 x+1)}{5 \times 7}$

$=\frac{(5 x+1)}{5} \times \frac{(7 x+1)}{7}$

$=\left(x+\frac{1}{5}\right)\left(x+\frac{1}{7}\right)$

Hence, factorisation of $x^{2}+\frac{12}{35} x+\frac{1}{35}$ is $\left(x+\frac{1}{5}\right)\left(x+\frac{1}{7}\right)$.