In the leaching method, bauxite ore is digested with a concentrated solution of NaOH that produces 'X'. When CO$$_2$$ gas is passed through the aqueous solution of 'X', a hydrated compound 'Y' is precipitated. 'X' and 'Y' respectively are:

We begin by recalling that bauxite, the principal aluminium ore, is essentially a hydrated oxide of aluminium and is conveniently written as $$\mathrm{Al_2O_3\cdot xH_2O}$$. In the Bayer leaching process this ore is digested with a concentrated solution of caustic soda.

First we need the chemical equation that describes what happens when bauxite meets hot concentrated NaOH. The aluminium oxide dissolves, forming a soluble complex called sodium aluminate while the iron and silica impurities remain undissolved.

$$\mathrm{Al_2O_3\cdot xH_2O + 2\,NaOH + (x\!-\!1)H_2O \longrightarrow 2\,Na[Al(OH)_4]}$$

All water molecules have been shown explicitly so that the balancing is perfectly clear. The product $$\mathrm{Na[Al(OH)_4]}$$ is the aqueous complex of aluminium; this is the substance labelled ‘X’ in the question.

Now the next operation in the Bayer cycle is to bubble carbon dioxide through the clear filtrate that contains ‘X’. The CO$$\_2$$ neutralises the alkaline solution, decomposing the aluminate ion and precipitating hydrated aluminium oxide (which you may also write as aluminium hydroxide, because the two notations represent the same solid).

$$\mathrm{2\,Na[Al(OH)_4] + CO_2 \longrightarrow 2\,Al(OH)_3\!\downarrow + Na_2CO_3 + H_2O}$$

The freshly precipitated gel $$\mathrm{Al(OH)_3}$$ always retains some water of hydration, so chemists very often prefer to write it in the compact form $$\mathrm{Al_2O_3\cdot xH_2O}$$. This hydrated oxide is the solid designated as ‘Y’ in the problem.

Summarising, we therefore identify

$$X = \mathrm{Na[Al(OH)_4]} \qquad\text{and}\qquad Y = \mathrm{Al_2O_3\cdot xH_2O}$$

Inspecting the given options, only Option D contains exactly these two formulas in the correct order.

Hence, the correct answer is Option 4.