Final answer:
The iliolumbar ligament is the one that connects L4, L5 to the sacrum and iliacus, thus providing stability to the lumbosacral region.
Step-by-step explanation:
The ligament that attaches L4, L5 to the sacrum and iliacus is the iliolumbar ligament. This ligament is important in stabilizing the lumbosacral junction where the lumbar spine meets the sacrum. It helps in supporting the fifth lumbar vertebra to the ilium, which is part of the hip bone. The options provided also include the sacroiliac ligament which is divided into anterior and posterior parts and supports the sacroiliac joint, the sacrotuberous ligament running from the sacrum to the ischial tuberosity, and the unrelated inguinal ligament, which is found at the anterior pelvis. However, the correct answer for the ligament that attaches specifically to the L4, L5 vertebrae, the sacrum, and iliacus is the iliolumbar ligament, making the answer A. Iliolumbar ligament.