Factorise: $12 x^{2}-7 x+1$

(A) To factorise the quadratic polynomial $12 x^{2}-7 x+1$,we use the splitting the middle term method.
Here,the coefficient of $x^{2}$ is $a = 12$,the coefficient of $x$ is $b = -7$,and the constant term is $c = 1$.
We need to find two numbers $l$ and $m$ such that their sum $l + m = b = -7$ and their product $lm = a \times c = 12 \times 1 = 12$.
The two numbers that satisfy these conditions are $-4$ and $-3$,since $(-4) + (-3) = -7$ and $(-4) \times (-3) = 12$.
Now,rewrite the middle term $-7x$ as $-4x - 3x$:
$12 x^{2} - 7 x + 1 = 12 x^{2} - 4 x - 3 x + 1$
Group the terms and factor out the common factors:
$= 4 x(3 x - 1) - 1(3 x - 1)$
Finally,factor out the common binomial $(3 x - 1)$:
$= (3 x - 1)(4 x - 1)$
Thus,the factors are $(3 x - 1)(4 x - 1)$.