[Noob Question] Having a basic, unchanging background spawn all around player.

I've looked everywhere for some way to have "infinite terrain". My background tiles are identical, and when I try to either move it when the player moves or make it a child to the player, the background seems to move and the player looks static. This is because it's constantly drawing the background where the player is.
I came up with a little script that, when the player passes the middle point of the starting background, inefficiently instantiates tons of objects. It looks like a plus shape, but has no diagonal direction, which makes it unusable and incomplete.

Is there some way to make the backgrounds spawn all around the player. without the ("static player") problem I described above?
Here's the "cross" part of the script, where terrainGrowRate = 1 and iLeft, iRight, etc... are all initialized to 0:

Code (CSharp):

1.         if (!(CrossPlatformInputManager.GetAxis ("Horizontal") == 0 && CrossPlatformInputManager.GetAxis ("Vertical") == 0)){
2.             //Left of middle
3.             if(background.position.x/2 > transform.position.x) {
4.                 Instantiate (background, background.position+new Vector3(terrainGrowRate*-1*iLeft,0f), transform.rotation);
5.                 iLeft+=1f;
6.  
7.             }
8.             //Right of middle
9.             if(background.position.x/2 < transform.position.x) {
10.                 Instantiate (background, background.position+new Vector3(terrainGrowRate*iRight,0f), transform.rotation);
11.                 iRight+=1f;
12.  
13.             }
14.             //Up of middle
15.             if(background.position.y/2 < transform.position.y) {
16.                 Instantiate (background, background.position+new Vector3(0f,terrainGrowRate*iUp), transform.rotation);
17.                 iUp+=1f;
18.             }
19.             //Down of middle
20.             if(background.position.y/2 > transform.position.y) {
21.                 Instantiate (background, background.position+new Vector3(0f,terrainGrowRate*-1*iDown), transform.rotation);
22.                 iDown+=1f;
23.  
24.             }
25.  
26.         }

As you can see (from my non-existent scripting ability), I'm new to Unity. Can anyone please help out? Thanks!

 

maybe using texture offset ?

Code (CSharp):

1. using System.Collections;
2. using System.Collections.Generic;
3. using UnityEngine;
4.  
5. public class tex : MonoBehaviour {
6.  
7.     Renderer rend;
8.     public float speed;
9.     Vector2 offset;
10.  
11.     void Start () {
12.         rend = GetComponent<Renderer>();
13.     }
14.  
15.     void Update () {
16.         float x = Input.GetAxis ("Horizontal");
17.         float y = Input.GetAxis ("Vertical");
18.         offset = new Vector2 (offset.x += x * Time.deltaTime * speed, offset.y += y * Time.deltaTime * speed);
19.         rend.material.SetTextureOffset("_MainTex", offset);
20.     }
21. }

here example 5.5.3