The numbers 42, 43, 44, 44, (2x + 3), 45, 45, 46, 47 have been arranged in an ascending order and their median is 45.

Arranging the given data in ascending order:
42, 43, 44, 44, (2x + 3), 45, 45, 46, 47

Number of terms = 9 (odd)

$\therefore$ Median $=\left(\frac{n+1}{2}\right)^{\text {th }}$ term

$\Rightarrow 45=\left(\frac{9+1}{2}\right)^{\text {th }}$ term

$\Rightarrow 45=(5)^{\text {th }}$ term

$\Rightarrow 45=2 x+3$

$\Rightarrow 2 x=45-3$

$\Rightarrow 2 x=42$

$\Rightarrow x=21$

Hence, the value of x is 21.

Arranging the given data in ascending order:
42, 43, 44, 44, 45, 45, 45, 46, 47

Here, 45 occurs maximum number of times.
∴ Mode = 45

Hence, the mode of the data is 45.