Short answer x = 10
Remark
An angle bisector divides (in this case) the base in the same ratio as the two sides.
Formula
8 / 12 = (x + 4)/(2x + 1)
Solve
Reduce the left hand side.
2/3 = (x + 4)/(2x + 1) Cross multiply
2*(2x + 1) = 3 * (x + 4) Remove the brackets
4x + 2 = 3x + 12 Subtract 3x from both sides
4x - 3x + 2 = 12 Subtract 2 from both sides
x = 12 - 2
x = 10