Where does the 4 come from?

I'm sure there is a more technically worded answer, but here I go.

That 4 comes from thinking: what can I do now to the left side of the equation so the x is more and more "alone"?

The x is DIVIDED by 4 right now. The way of making it disappear is to MULTIPLY it by 4.

That's where the 4 came from.

(Whatever you do to the left side, you also have to do it to the right side, so both sides are multiplied by that 4)

Is exactly the same as asking "where did that 5 came from?" in the previous step.

You just needed to make the -5 disappear. So the way of making it disappear is to ADD a 5. So you add a 5 to both sides and the -5 then disappears.
it doesn't need to "come from" anywhere; as long as you do an operation to **both sides of the equals sign**, the equality is maintained.

In order to solve for a variable x, which is done by getting it alone on one side of the equals sign (i.e. 'isolate x'), you have to do the opposite of what is currently being done to it: for example when we have 5 being added to x in the equation x + 5 = 3, we do the opposite by subtracting 5 from both sides of the equation to get x = 3 - 5 = -2.

If x is being divided by 4 as in the equation x/4 = 2, we do the opposite of division and multiply  both sides by 4 to get x = 2\*4 = 8. Remember that a fraction is just a division!

When multiple operations are being done to x, to make it easier for yourself *you go the reverse of the order of operations*: first "undo" the additions / subtractions, then "undo" the multiplications/divisions, then "undo" the powers. In your equation, we have:

1. Undo the subtraction of 5 by adding 5 to both sides: (x\^2)/4 - 5 = 4 -> (x\^2)/4 = 9
2. Undo the division by 4 by multiplying both sides by 4: (x\^2)/4 = 9 -> x\^2 = 36 (remember, **a fraction is just a division**, so you undo it via multiplication!)
3. Undo the power of 2 by taking the square root of both sides: x\^2 = 36 -> x = $+ or -$ 6

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Edit: From your other comment it's clear that the context of this is quadratics; note that an equation is usually only solvable this was if you only have one power of x in play, i.e. only x\^2 in the equation; if both x\^2 and x (and/or other powers) are present x can't be isolated via this simple means, so more advanced methods like factoring, "completing the square", and the quadratic formula must be used. The quadratic formula is the "x=-b+-$b2-4ac$ / 2a" you mention and you could certainly use it to solve this by taking a = 1/4, b = 0, c = -1, but since there is no x term (only x\^2) this would be an overcomplication.
The objective is to isolate the variable. Basically get x alone on one side of the equal sign. You’re dividing by four so you do the opposite  to get rid of it. Multiplying by four does that to leave just x squared on the left hand side
You are multiplying both side of the equation with 4 to cancel out the 4 on the left side.
If you know how to solve linear equations, it’s the same process, you just have to pretend you’re solving for x^2 instead of x. Then you get something like x^2 = c where you know what value c is. So you can take the square root of both sides to get x, as shown in the last step in the photo.

If you don’t know how to solve linear equations, put away that book and learn to solve linear equations.
The four is in the denominator of the fraction which implies division is currently taking place.  You simply DO the inverse operation of what you SEE. So you SEE division so then you must DO multiplication.
This is why I always preferred to move parts of the equation from one side to the other. But I'm a visual learner so it made sense.
A maths teacher as a kid taught me "change side change sign"
Neutralizing the denominator
To eliminate the denominator you must multiply by the same value as the denominator.
Multiply 4 on both sides to get rid of the fraction.
So (×^2/4) can be x^2