If the semi-perimeter and area of a rectangular field whose length and breadth are 'x' and 'y' is 12cm and 28cm
2
, respectively, then find the value of x
4
+x
2
y
2
+y
4
.

Given parameters:
Length = x, Breadth = y

Semi-perimeter =x+y=12

Area =xy=28

We need to evaluate the expression: x
4
+x
2
y
2
+y
4
.
We can factorize or rewrite this expression using algebraic identities:

x
4
+x
2
y
2
+y
4
=(x
2
+y
2
)
2
−x
2
y
2

First, let's find (x
2
+y
2
):

x
2
+y
2
=(x+y)
2
−2xy
Substitute the given values:

x
2
+y
2
=(12)
2
−2(28)
x
2
+y
2
=144−56=88
Now, substitute this back into the rewritten target expression:

x
4
+x
2
y
2
+y
4
=(x
2
+y
2
)
2
−(xy)
2

=(88)
2
−(28)
2

Using the identity a
2
−b
2
=(a−b)(a+b):

=(88−28)(88+28)
=(60)×(116)
=6960