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How do I determine how many spell slots I have if I multiclass into more than one spellcasting class? For example, I have a character that is a level 3 Wizard, a level 5 Eldritch Knight Fighter, a level 4 Arcane Trickster Rogue, a level 4 Paladin, and a 4 Warlock. How many spell slots do I have and how do I determine this?

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You would use the rules for Multiclass Spellcasting.

The Multiclass Spellcasting rules state (PHB, p. 164-165):

You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.

The multiclassing rules for the artificer class are not included there, since the class was not published in the PHB. However, the artificer class description includes an "Optional Rule: Multiclassing" section, which states (E:RftLW, p. 54; WGtE, p. 176):

Spell Slots. Add half your levels (rounded up) in the artificer class to the appropriate levels from other classes to determine your available spell slots.

So in the case of your example, your spellcaster level is 8. Therefore, you would have four 1st-level spell slots, three 2nd-level slots, three 3rd-level slots, and two 4th-level slots.

I got eight from 3 levels in wizard + ((5 levels in Fighter with Eldritch Knight + 4 levels in Rogue with Arcane Trickster) over 3) + (4 levels in Paladin over 2). That's 3 + (9/3) + (4/2) or 3 + 3 + 2.

The Multiclass Spellcaster table is reproduced below:

\begin{array}{c|ccc ccc ccc } \text{Level}& \text{1st}& \text{2nd}& \text{3rd}& \text{4th}& \text{5th}& \text{6th}& \text{7th}& \text{8th}& \text{9th} \\ \hline \text{1st}&2&-&-&-&-&-&-&-&- \\ \text{2nd}&3&-&-&-&-&-&-&-&- \\ \text{3rd}&4&2&-&-&-&-&-&-&- \\ \text{4th}&4&3&-&-&-&-&-&-&- \\ \text{5th}&4&3&2&-&-&-&-&-&- \\ \text{6th}&4&3&3&-&-&-&-&-&- \\ \text{7th}&4&3&3&1&-&-&-&-&- \\ \text{8th}&4&3&3&2&-&-&-&-&- \\ \text{9th}&4&3&3&3&1&-&-&-&- \\ \text{10th}&4&3&3&3&2&-&-&-&- \\ \text{11th}&4&3&3&3&2&1&-&-&- \\ \text{12th}&4&3&3&3&2&1&-&-&- \\ \text{13th}&4&3&3&3&2&1&1&-&- \\ \text{14th}&4&3&3&3&2&1&1&-&- \\ \text{15th}&4&3&3&3&2&1&1&1&- \\ \text{16th}&4&3&3&3&2&1&1&1&- \\ \text{17th}&4&3&3&3&2&1&1&1&1 \\ \text{18th}&4&3&3&3&3&1&1&1&1 \\ \text{19th}&4&3&3&3&3&2&1&1&1 \\ \text{20th}&4&3&3&3&3&2&2&1&1 \\ \end{array}

Thus far, all of our spell slots are essentially combined into a single shared pool of slots. None of them are class-specific, but they can be used by all of your classes. You regain these spell slots on a long rest according to each of the Spellcasting features. This does still allow you to use features like the Wizard's Arcane Recovery to gain some back on a short rest (in this case, either one 2nd-level slot or two 1st-level slots).

Also note that this does not change what spells you can learn/know or have prepared; that is determined as if you were a single-classed member of each of those classes. In other words, you separately add spells to your spellbook (and prepare spells from it) as a level 3 wizard, have the known spells of a level 5 Eldritch Knight, have the known spells of a level 4 Arcane Trickster, prepare spells as a level 4 Paladin, and have the known spells of a level 4 Warlock. See this Q&A for more details: If I multiclass into 2 or more spellcasting classes, how do I determine my known/prepared spells?

Notice that you add the Fighter and Rogue levels before dividing them. Otherwise you would be a level 7 spellcaster. (Opinions vary on the order of operations here; see this Q&A for more details: How do paladin and ranger class levels add up for multiclass spellcasting?)
Also, if you were less than level 3 in either Fighter or Rogue, you would not have the subclass that grants you the Spellcasting feature, making you a level 6 spellcaster. The same applies if you were only level 1 in Paladin.

But what about my Warlock levels?

The same section on Multiclass Spellcasting also says this about Pact Magic:

If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.

What this essentially means is that these slots are completely independent of the rest. They are not affected by the rest of your levels. You can however, still cast your Warlock spells with the slots from your Wizard, Fighter, Rogue, and Paladin classes and you can use your Warlock spell slots (which recover on a short rest!) to cast your spells from those classes. They're separate but compatible.

So your total number of spell slots, after adding in your 4 levels in Warlock (which according to the Warlock Table gives you 2 second level spell slots) are 4 first level spell slots, 5 second level slots, 3 third level slots, and 2 fourth level slots. And two of those second level slots recover on a short rest.

Special note: Eldritch Smite is only works with Warlock slots

András originally posted the following answer to the Q&A Can a multiclassed warlock expend a Spellcasting spell slot of a non-warlock class to use the Eldritch Smite invocation?, but I wanted to add it here for the sake of completeness:

Only warlock spell slots from the Pact Magic feature can be expended to use Eldritch Smite; spell slots from the Spellcasting feature do not work with it.

An unofficial tweet by rules designer Jeremy Crawford in November 2017 confirms that this was intentional (unlike the paladin's Divine Smite, which had similar wording initially but was changed in errata later to work with any spell slot):

Talking about Eldritch Smite; can you only use Warlock spell slots to power it, or can you use any slot (à la Divine Smite)? The RAW on it specifies Warlock slots.

Eldritch Smite works with warlock spell slots only—the ones you get from Pact Magic.

Judging by the similar wording, we can assume the same applies to the following other Eldritch Invocations: Bewitching Whispers, Dreadful Word, Minions of Chaos, Sculptor of Flesh, Sign of Ill Omen, and Thief of Five Fates.

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