Do you think the equations (x−1)(x+3)=17+x and (x−1)(x+3)+500=517+x should have the same solution set? why?
do you think the equations (x−1)(x+3)=17+x and 3(x−1)(x+3)=51+3x should have the same solution set? explain why.

Those two pair of equations have the same solution set.

Step-by-step explanation:

There are two equations  

(x-1)(x+3)=17+x (1) and  

(x-1)(x+3)+500=517+x (2)

We have to check the same solution set will be there for equations (1) and (2) or not.

Now, we are going to rearrange the equation (2).

(x-1)(x+3)+500=517+x

⇒ (x-1)(x+3)=517-500+x

⇒(x-1)(x+3)=17+x

This is the same equation as equation (1).  

Therefore, there will be the same solution set for equations (1) and (2).  (Answer)

There are two equations  

(x-1)(x+3)=17+x (3) and  

3(x-1)(x+3)+500=51+3x (4)

We have to check the same solution set will be there for equations (3) and (4) or not.

Now, we are going to rearrange the equation (4).

3(x-1)(x+3)+500=51+3x

⇒ 3(x-1)(x+3)=3(17+x)

⇒(x-1)(x+3)=17+x

This is the same equation as equation (3).  

Therefore, there will be the same solution set for equations (3) and (4). (Answer)