Theory:

Identity \(2\): (a+b)2=a2+2ab+b2
 
Let us construct a figure of four regions. The two square shaped regions with the dimensions of \(3 × 3\) (Blue) and \(2 × 2\) (Yellow). Observe the remaining two rectangle shaped regions. Both are in \(3 × 2\) (Green) dimension.
 
pic 4.png
 
By observing the above rectangle, we can notice that:
 
\(\text{Area of the bigger square = Area of the two small square + Area of the two rectangles}\)
 
\(3 + 2\)\(2 + 3\) \(=\) \((3 × 3) + (2 × 3) + (3 × 3) + (2 × 2)\)
 
Now, we simplify the LHS and RHS of the above expression.
 
LHS \(=\) \(3 + 2\)\(2 + 3\) \(=\) \(5×5 = 25\)
 
RHS \(=\) \((3 × 3) + (2 × 3) + (3 × 3) + (2 × 2)\)
 
         \(=\) 32+(2×3)+(3×2)+22 \(=\) \(9+6+6+4 = 25\)
 
Therefore, LHS \(=\) RHS
 
Similarly, if we use the variables in this case instead of number we get:
 
pic 5.png
 
Assume the square of ABCD of side \(a + b\). From the above figure, we can get that:
 
\(\text{The total area of the bigger square = The area of the two small squares × The are of the two rectangles}\)
 
That is, \((a + b)^2\)=a2+ab+ba+b2
 
Since, \(ba=ab\);  \((a + b)^2\) =a2+ab+ab+b2=a2+2ab+b2.
 
Therefore, (a+b)2=a2+2ab+b2 is a identity.
Example:
Simplify the expression (x+2)(x+2) using the identity (a+b)2=a2+2ab+b2.
 
Now write the given expression (x+2)(x+2) with respect to the given identity (a+b)2=a2+2ab+b2.
 
(a+b)(a+b)=(x+2)(x+2)(x+2)2=(x2+2(2×2)x+22)
 
Now simplify the expression.
 
=(x2+8x+4).
 
Therefore, (x+2)(x+2)=(x2+8x+4).