Nomenclature of multiple double bonds?
The multiplying affixes are given by Table 11 of the Blue Book, referenced in R-0.1.4 and R-4.1:
Number  Numerical term | Number  Numerical term
1       mono-          | 90      nonaconta-
2       di-            | 100     hecta-
3       tri-           | 200     dicta-
4       tetra-         | 300     tricta-
5       penta-         | 400     tetracta-
6       hexa-          | 500     pentacta-
7       hepta-         | 600     hexacta-
8       octa-          | 700     heptacta-
9       nona-          | 800     octacta-
10      deca-          | 900     nonacta-
11      undeca-        | 1000    kilia-
12      dodeca-        | 2000    dilia-
20      icosa-         | 3000    trilia-
30      triaconta-     | 4000    tetralia-
40      tetraconta-    | 5000    pentalia-
50      pentaconta-    | 6000    hexalia-
60      hexaconta-     | 7000    heptalia-
70      heptaconta-    | 8000    octalia-
80      octaconta-     | 9000    nonalia-

Therefore, the answer to your question is 2,4,6,8-tetraene.

Extra information
Note that there is no elision of vowels, specified by R-0.1.7:

R-0.1.7.1 Vowels are systematically elided as follows:
(a) the terminal &quot;e&quot; in names of parent hydrides when followed by a suffix beginning with &quot;a&quot;, &quot;i&quot;, &quot;o&quot;, &quot;u&quot;, or &quot;y&quot;;
(b) in the Hantzsch-Widman system, the final &quot;a&quot; of an element prefix when followed by a vowel;
(c) the terminal &quot;a&quot; in the names of numerical multiplicative affixes when followed by a suffix beginning with &quot;a&quot; or &quot;o&quot;, or a Hantzsch-Widman prefix or stem beginning with a vowel;
(d) the terminal &quot;a&quot; of an element prefix in &quot;ababa&quot; repeating unit names (see R-2.3.3.2) and the terminal &quot;o&quot; of a replacement infix when followed by a vowel;

The relevant part is (c), which states that the final &quot;a&quot; in the above affixes are elided when the next letter is &quot;a&quot; or &quot;o&quot;, but in this case the next letter is &quot;e&quot; so it is not elided.
The following are given examples of elision:

benzenehexaol &gt; benzenehexol
[1,1'-binaphthalene]-3,3',4,4'-tetraamine &gt; [1,1'-binaphthalene]-3,3',4,4'-tetramine
1,3,5,7-tetraoxaocane &gt; 1,3,5,7-tetraoxocane