Tail Recursion: Difficult Solutions

code writing

Write in a tail recursive function that takes in a list and sorts it quickly. You may use the append function, which takes in two lists and puts them together. Hint: Write some helper functions to help you break down lst into three lists: values less than the car of lst, values equal to the car, and values greater than the car.
(define (quicksort-tail-recursive lst)
    (define (helper lst greater equal less)
        (if (null? lst)
            nil
            (append (quicksort-tail-recursive (remove-less (cdr lst) (car lst) nil))
                (append (remove-equal (cdr lst) (car lst) (list (car lst))) (quicksort-tail-recursive (remove-greater (cdr lst) (car lst) nil))))))

(define (remove-less lst x less)
    (cond ((null? lst) less)
    ((< x (car lst)) (remove-less (cdr lst) x (cons (car lst) less))
    (else (remove-less (cdr lst) x less))))

(define (remove-greater lst x greater)
    (cond ((null? lst) greater)
    ((> x (car lst)) (remove-greater (cdr lst) x (cons (car lst) greater))
    (else (remove-greater (cdr lst) x greater))))

(define (remove-equal lst x equal)
    (cond ((null? lst) equal)
    ((> x (car lst)) (remove-equal (cdr lst) x (cons (car lst) equal))
    (else (remove-equal (cdr lst) x equal))))