Explanation

(a)
$\frac{3x^{2} + 1}{x^{2} - 6x + 8} = 3 + \frac{18x - 23}{x^{2} - 6x + 8}$ [On dividing] ...(i)
Now, $\frac{18x - 23}{(x - 2)(x - 4)} = \frac{A}{(x - 2)} + \frac{B}{(x - 4)}$
$\Rightarrow \ \ \ 18x - 23 = A(x - 4) + B(x - 2)$
$\Rightarrow \ \ \ 18x - 23 = (A + B)x - 4A - 2B$
On equating the coefficient of $x$ and constant term, we get
$A + B = 18$
And $- 4A - 2B = - 23$
On solving these equations, we get
$A = - \frac{13}{2},\ \ B = \frac{49}{2}$
$\therefore\ \ \ \ \ \frac{18x - 23}{(x - 2)(x - 4)} = - \frac{13}{2(x - 2)} + \frac{49}{2(x - 4)}$
Then, from Eq. (i), we get
$\frac{3x^{2} + 1}{x^{2} - 6x + 8}$
$= 3 - \frac{13}{2(x - 2)} + \frac{49}{2(x - 4)}$