Why is it called synthetic division?

Umm, maybe because it is not "real" division, it is fake, or "synthetic", division.

[In building materials, "synthetic" signifies "not natural".]

[Things like this, definitions, I mean, you can search Google or Wikipedia or Yahoo.]

It is difficult to show here the form or layout of the synthetic division, but let me try.

"divide by (x+3)" means (x+3) may be a factor, and so (-3) may be a root, of the expression (2x^5 +3x^4 +25x^2 -1).

So, (-3) is the first divisor of sort.

-3...|2.....3.....0.....25.....0.....-1

............-6.....9....-27.....6....-18

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........2..-3.....9.....-2......6....-19

It is:

a) Bring down 2

b) -3*2 = -6

c) 3 +(-6) = -3

d) -3 *(-3) = 9

e) 0 +9 = 9

f) -3 *9 = -27

g) 25 +(-27) = -2

h) -3 *(-2) = 6

i) 0 +6 = 6

j) -3 *6 = -18

k) -1 +(-18) = -19

So the quotient is 2x^4 -3x^3 +9x^2 -2x +6, remainder (-19).

If you studied synthetic division, you may understand the above.

Not good to explain it here.