I think 19 is the correct answer if you approach this as a MATH problem:
1+4=5
2+5=12
3+6=21
8+11=19
Because of course, 8+11=19 and always will, independent of the poor notation and mathematical mistakes above it.

However, it is more fun to approach this as a LOGIC problem and for that I can come up with 5 different solutions, all of which are "correct" and many of you have suggested them. First, lets write this as a logic statement:

IF
(1,4,5)
(2,5,12)
(3,6,21)
THEN
(8,11,?)

The first, and probably simplest solution, is probably to add the first column to the product of the second column multiplied by the line number to produce the third:
Line 1 - (1,4,5)
Line 2 - (2,5,12)
Line 3 - (3,6,21)
Line 4 - (8,11,52)

The second solution is to add the third column of the previous line to the first two columns of the current line:
(1,4,5)
(2,5,12)
(3,6,21)
(8,11,40)

The third is to add the first column to the product of the first and second:
(1,4,5)
(2,5,12)
(3,6,21)
(8,11,96)

The fourth is to recognize and complete the logical progression of 1,2,3... and 4,5,6.., until you reach 8 and 11, then add the first column to the product of the second column and first column (or line number) to produce the third:
Line 1 - (1,4,5)
Line 2 - (2,5,12)
Line 3 - (3,6,21)
Line 4 - (4,7, 32)
Line 5 - (5,8, 45)
Line 6 - (6,9, 60)
Line 7 - (7,10, 77)
Line 8 - (8,11,96)

The last that I can come up with is to recognize and complete the logical progression of 1,2,3... and 4,5,6.., until you reach 8 and 11, then add the third column of the previous line to the first two columns of the current line (similar to the second solution but completing the progression):
(1,4,5)
(2,5,12)
(3,6,21)
(4,7, 32)
(5,8, 45)
(6,9, 60)
(7,10, 77)
(8,11, 96)

As you can see there are three ways to reach 96 (yes, 3 and 4 are basically the same but with a slightly different thought process). And I withdraw my previous 116 solution because of an addition mistake! Thanks OP for posting!

Last edited by Johnsclist; 09/21/21.