How to make my enemy discover, chase and lose player.

Sorry my answer wasn't detailed enough because I didn't understand your question clearly. I have rewritten your code with performance and readability in mind, and tried to explain everything using comments. Btw <= means less than or equal.

 using UnityEngine;
 
 public class GuardMovement : MonoBehaviour
 {
     public float playerDetectionRange = 50;
     public float playerDeathZone = 1f;
     public float patrolDistance = 5f;
     public float speed = 1f;
     // Using a multiplyer to avoid having another var
     // for walking speed and anothor one for current speed.
     public float runningSpeedMultiplyer = 1.5f;
 
     Animator animator;
 
     // Patroling
     Vector3[] patrolPositions;
     int patrolPositionsIndex;
 
     Vector3 startPosition;
     Vector3 targetPosition;
     Transform player;
     bool isTargetingPlayer = false;
 
     void Start()
     {
         animator = GetComponent<Animator>();
 
         startPosition = transform.position;
 
         patrolPositions = new Vector3[] {
             new Vector3(startPosition.x + patrolDistance, startPosition.y, startPosition.z),
             new Vector3(startPosition.x - patrolDistance, startPosition.y, startPosition.z)
         };
 
         // To save preformance if your game has only 1 player,
         // then change it in the declearation to be:
         // static Transform player;
         // and add that here:
         // if (player == null)    
         player = GameObject.FindGameObjectWithTag("Player").transform;
     }
 
     void Update()
     {
         if (IsPlayerInDetectionRange())
         {
             // The player is moving so we need to update the every frame.
             SetTargetPosition(player.position);
 
             // Will only not target the player while in range
             // once he enters the detection range.
             if (!isTargetingPlayer)
             {
                 isTargetingPlayer = true;
                 OnPlayerEnter();
             }
 
             if (IsPlayerInAttackRange())
             {
                 // Do whatever you want in here to
                 // indicate that the player is dead.
 
                 // player.Kill();
             }
         }
         // If not in range.
         else
         {
             // will only taget the player while not in range
             // once he leaves the detection range.
             if (isTargetingPlayer)
             {
                 isTargetingPlayer = false;
                 OnPlayerLeave();
             }
 
             // Patrol only while no player in range.
             Patrol();
         }
 
         MoveToTargetPosition();
     }
 
     bool IsPlayerInDetectionRange()
     {
         if (Vector2.Distance(transform.position, player.position) < playerDetectionRange)
         {
             return true;
         }
         else
         {
             return false;
         }
     }
 
     bool IsPlayerInAttackRange()
     {
         if (Vector2.Distance(transform.position, player.position) < playerDeathZone)
         {
             return true;
         }
         else
         {
             return false;
         }
     }
 
     void OnPlayerEnter()
     {
         speed *= runningSpeedMultiplyer; // same as (speed = speed * runningSpeedMultiplyer)
     }
 
     void OnPlayerLeave()
     {
         speed /= runningSpeedMultiplyer; // same as (speed = speed / runningSpeedMultiplyer)
     }
 
     void Patrol()
     {
         // Putting it in a var just to make it easier to read.
         Vector3 currentPatrolPosition = patrolPositions[patrolPositionsIndex];
 
         // If we are not patroling (if after chasing a player)
         if (targetPosition != currentPatrolPosition)
         {
             SetTargetPosition(currentPatrolPosition);
         }
 
         // If we have reached our target patrol, then switch to the next one.
         if (transform.position == currentPatrolPosition)
         {
             patrolPositionsIndex++; // same as (patrolPositionsIndex = patrolPositionsIndex + 1)
 
             // We currently have 2 patrol position stored in an array,
             // if the index is bigger than or equal to the length of the array (2),
             // then we will start from the beggining (index 0).
             if (patrolPositionsIndex >= patrolPositions.Length)
                 patrolPositionsIndex = 0;
 
             SetTargetPosition(patrolPositions[patrolPositionsIndex]);
         }
     }
 
     void SetTargetPosition(Vector3 newTargetPosition)
     {
         targetPosition = newTargetPosition;
 
         // if is going left
         if (transform.position.x < targetPosition.x)
         {
             animator.SetInteger("Direction", 3);
         }
         // if is going right
         else
         {
             animator.SetInteger("Direction", 1);
         }
     }
 
     void MoveToTargetPosition()
     {
         transform.position = Vector3.MoveTowards(transform.position, targetPosition, speed * Time.deltaTime);
     }
 }