Can someone help me with number 4 and 5? Thank you very much. I attached an image of the problems. The book is brand new and i cant find any help.

You seem to have misunderstood the purpose of this board. Show us what you have tried and where you have problems and we will attempt to assist you and give hints.

I will say this much- problems 4 and 5 do NOT ask you to solve the equation. They only ask you to show that a given function is a solution. If you were given the equation $x^4- 3x^2+ 4x- 12= 0$, and asked to show that x= 2 is a solution, you would NOT attempt to solve the equation- just replace x with the number "2" and see if it works.

im sorry, i should say what i am doing. i was going to take the 2nd partial of (x+t)^4 with respect to x and t. then, see if they are equal. is this what it wants me to do?

I ended up getting 12(x+t)^2 for both, so they are equal. That was easy...sorry. now for b.