Hello and welcome to Science Buddies!
That's an excellent question regarding a seemingly confusing name.
Let's first think about what solubility means, because that's the easier concept of the two. When something - let's say, table salt (NaCl) - "
dissolves" in water, this is due to
polarity. Water is a polar molecule, with a positive end towards the hydrogens and a negative end towards the oxygen. Salt is also composed of polar components: a sodium ion (Na+) and a chloride ion (Cl-). Outside of water, these ions are attracted strongly to each other, making a single solid compound. But water breaks them down into their ions by attracting and surrounding them, pulling them apart - pointing their negative oxygens towards the positive Na+ and their positive hydrogens towards the negative Cl-.
This is the basic principle of solubility in water. If it is polar or made of polar components, water will "break it down" and dissolve it.
Note that dissolving something does not actually cause a chemical reaction; in other words, neither the water nor the salt are chemically changed into something new.
Now for the other one. "Hydrolyzed" refers to a chemical's ability to undergo
hydrolysis, a special type of chemical reaction that involves water as a reactant. Here, water turns from H2O into a hydrogen ion (H+) and a hydroxide ion (OH-). Then, the hydrolyzed substance - like starch - will break a bond to take the H+ ion.
Note that here, both the water and the starch have been involved in a reaction that changes both of their structures. Water has been "ionized" into two different things that aren't water, and the starch has had a chemical bond actually broken.
In starch specifically, the hydrolysis reaction will break it down. Starch is a polysaccharide, which just means it's a bunch of molecules called "saccharides" strung together on a series of chemical bonds. Hydrolysis breaks these bonds down, cutting starch into smaller and smaller strings of saccharides. Think of it like a string of Christmas lights: each light is a saccharide, and the water is taking a pair of scissors to the wire between them. (Note: do not try this at home.

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Hope that helps! Feel free to ask more questions!