Expanding and Simplifying the Expression (a+b+c+d)^2+(a+bcd)^2+(a+cbd)^2+(a+dbc)^2
This expression represents the sum of squares of four different terms, each involving the variables 'a', 'b', 'c', and 'd'. Let's break down how to simplify this expression and uncover its underlying pattern.
Expanding the Squares
First, we need to expand each of the squares using the FOIL (First, Outer, Inner, Last) method or simply by applying the algebraic identity: (x + y)^2 = x^2 + 2xy + y^2

(a+b+c+d)^2: This expansion results in a large number of terms, but we can use a systematic approach to avoid missing any:
 Square the first term: a^2
 Multiply the first term by the second term and double the result: 2ab
 Multiply the first term by the third term and double the result: 2ac
 Multiply the first term by the fourth term and double the result: 2ad
 Square the second term: b^2
 Multiply the second term by the third term and double the result: 2bc
 Multiply the second term by the fourth term and double the result: 2bd
 Square the third term: c^2
 Multiply the third term by the fourth term and double the result: 2cd
 Square the fourth term: d^2
 Combine all terms: a^2 + 2ab + 2ac + 2ad + b^2 + 2bc + 2bd + c^2 + 2cd + d^2

(a+bcd)^2: Following the same method as above, we get:
 a^2 + 2ab  2ac  2ad + b^2  2bc  2bd + c^2 + 2cd + d^2

(a+cbd)^2:
 a^2  2ab + 2ac  2ad + b^2  2bc + 2bd + c^2  2cd + d^2

(a+dbc)^2:
 a^2  2ab  2ac + 2ad + b^2 + 2bc  2bd + c^2 + 2cd + d^2
Simplifying the Expression
Now, we can add all the expanded terms together. Notice that many terms cancel out due to their opposite signs:
 a^2: (4)a^2
 b^2: (4)b^2
 c^2: (4)c^2
 d^2: (4)d^2
 2ab: (2)2ab  (2)2ab = 0
 2ac: (2)2ac  (2)2ac = 0
 2ad: (2)2ad  (2)2ad = 0
 2bc: (2)2bc  (2)2bc = 0
 2bd: (2)2bd  (2)2bd = 0
 2cd: (2)2cd  (2)2cd = 0
The simplified expression is: 4a^2 + 4b^2 + 4c^2 + 4d^2
Final Result
We can further simplify by factoring out 4: 4(a^2 + b^2 + c^2 + d^2)
Therefore, the simplified form of the original expression is 4(a^2 + b^2 + c^2 + d^2). This demonstrates that the expression represents four times the sum of squares of the individual variables.