Factorize the given expression 1 + 2ab - ({a^2} + {b^2}) .

To factorize the given expression 1 + 2ab - (a^2 + b^2), follow these steps:

Step 1: Distribute the negative sign inside the parentheses.

1 + 2ab - a^2 - b^2

Now the expression is:

1 + 2ab - a^2 - b^2

Step 2: Rearrange the terms to group the like terms together:

a^2 - 2ab + b^2 + 1
Step 3: Notice that a^2 - 2ab + b^2 is a perfect square trinomial, which can be factored as:

(a - b)^2

So, the expression becomes:

(a - b)^2 + 1

Step 4: The factored form of the given expression is:

(a - b)^2 + 1

Thus, the factorized form of 1 + 2ab - (a^2 + b^2) is (a - b)^2 + 1.