Simply convert $(34.4)_8$ and $(23.4)_8$ to decimal.
$(34.4)_8 = 28.5$ in decimal and $(23.4)_8 = 19.5$ in decimal.
$28.5 \times 19.5 = 555.75$
Now convert $555.75$ back to octal which is $(1053.6)_8$
$(\overset{\leftarrow}{34}.\overset{\rightarrow}4)_8$ to decimal
$\quad = 3 \times 8^1 + 4 \times 8^0 + 4\times 8^{-1}$
$\quad = 24 + 4 + 0.5$
$\quad = (28.5)_{10}$
$(\overset{\leftarrow}{23}.\overset{\rightarrow}4)_8$ to decimal
$\quad = 2 \times 8^1 + 3 \times 8^0 + 4\times 8^{-1}$
$\quad = 16 + 3 + 0.5$
$\quad = (19.5)_{10}$
$(28.5)_{10} \times (19.5)_{10} = (555.75)_{10}$
Now, $$(555.75)_{10} = (?)_8$$
To convert the integer part
$$\begin{array}{r|lcc}
8 & 555 & \\\hline
8&69 &3\\\hline
8&8 &5\\\hline
8&1& 0\\\hline
8&0& 1 & {\uparrow}
\end{array}$$
We get $1053.$
To convert the decimal, keep multiplying by $8$ till decimal part becomes $0.$
$0.75 \times 8 \to \underbrace{6}_{\text{keep the integral part}}.\underbrace{00}_{0 \text{ decimal part}}$
$$\therefore (555.75)_{10} = (1053.6)_8$$
Correct Answer: $A$