If a+b=2 and ab=4, then find the value of a
4
+b
4
+ab
3
+ba
3
.

We are given:
a+b=2
ab=4

We need to find the value of:

E=a
4
+b
4
+ab
3
+ba
3

Rearranging the terms:

E=(a
4
+b
4
)+ab(b
2
+a
2
)
E=(a
4
+b
4
)+ab(a
2
+b
2
)
First, let's find (a
2
+b
2
):

a
2
+b
2
=(a+b)
2
−2ab
a
2
+b
2
=(2)
2
−2(4)=4−8=−4
Now, let's find (a
4
+b
4
):

a
4
+b
4
=(a
2
+b
2
)
2
−2(ab)
2

a
4
+b
4
=(−4)
2
−2(4)
2
=16−2(16)=16−32=−16
Substitute these values back into the expression for E:

E=(−16)+4(−4)
E=−16−16=−32