(defun p* (p1 p2)
  "Multiplies the two given polynomials together."
    (mapcan #'(lambda (x)
      (mapcan #'(lambda (y)
        (list (cons (* (car x) (car y)) (concatenate 'list (cdr x) (cdr y))))
        ) p1)
    ) p2)
  )

(defun merge-variables (input &optional output)
  "Recursively collects terms given the original input"
  ;(print input)
  ;(print output)
  (if (null input)
    (return-from merge-variables output)) ; input is exhausted, so return output
  (merge-variables
        (mapcan #'(lambda (x)
          (if (not (equal (caar input) (car x)))
              (list x) )
        ) input) ; new input list will contain all variables not equal to the current variable.

        (check-for-zero-exponential (list (caar input)
          (apply #'+ (mapcan #'(lambda (x)
            (if (equal (caar input) (car x))
             (cdr x))
        ) input))) output) ; current variable (with accumulated exponents) will
                       ; be appended to the output list (providing its exponent is non-zero).
   )
)

(defun check-for-zero-exponential (variable-with-expo output)
  "If exponent is non-zero, append to output. Otherwise, returns the output"
  (if (eq 0 (cadr variable-with-expo))
    (return-from check-for-zero-exponential output)
    (return-from check-for-zero-exponential (concatenate 'list output (list variable-with-expo))))
  )

(defun evaluate-polynomial (p)
  (mapcar #'(lambda (x)
       ;(print x)
       (cons (car x) (merge-variables (cdr x))) ; cons original coefficient onto simplified list of variables for that term
    ) p)
  )

(equal '((4 (X 5) (X 2)) (4 (X 5) (Y 1))) (p* '((2 (X 2)) (2 (Y 1))) '((2 (X 5))))) ; evaluates to true
(evaluate-polynomial (p* '((2 (X 2)) (2 (Y 1))) '((2 (X 5))))) ; evalutes to ((4 (X 7)) (4 (X 7) (Y 1))) (incorrect)
(evaluate-polynomial '((4 (X 5) (X 2)) (4 (X 5) (Y 1)))) ; evaluates to ((4 (X 7)) (4 (X 5) (Y 1))) (correct)