QUESTION

# Evaluate the following equation  ${a^2} - {b^2} - {({a^{}} + b)^2}$

Hint: Here, we will be proceeding by firstly opening all the square brackets given in the equation. For opening the brackets, we need to know the basic identities. After opening all the brackets, we will have to do the simple mathematics. After this, we will get our required answer.

Given, ${a^2} - {b^2} - {(a + b)^2}$
Firstly, we need to open all the squares and brackets.
After opening the squares, the equation becomes
${a^2} - {b^2} - ({a^2} + 2ab + {b^2})$
Because ${(a + b)^2} = {a^2} + 2ab + {b^2}$
And after opening the brackets, the equation becomes
${a^2} - {b^2} - {a^2} - 2ab - {b^2}$
Now, it is the time to do the simple mathematics
After doing simple addition, the equation becomes
$- {b^2} - 2ab - {b^2}$
After simplifying, the equation becomes
$\Rightarrow - 2{{\text{b}}^2} - 2ab = - 2b(b + a)$
The above equation can also be written as
$- 2b(a + b)$