Factorization of 9x²+6xy+y²
Factors are numbers or expressions that we can multiply together to get a certain result. In order to factorise the given expression 9x²+6xy+y², we need to identify a suitable algebraic identity that can help us break down the expression into simpler factors.

Using the Perfect Square Trinomial Identity
One of the identities that we can use to factorise this expression is the Perfect Square Trinomial Identity, which states that (a+b)² = a² + 2ab + b². By comparing the given expression with this identity, we can see that it closely resembles the form of (a+b)².

Applying the Perfect Square Trinomial Identity
In this case, we can rewrite the given expression as (3x)² + 2(3x)(y) + (y)². This allows us to identify that the expression fits the form of (a+b)², where a = 3x and b = y.

Factorising the Expression
Now, we can factorise the expression as (3x + y)². This is the final factorised form of the given expression 9x²+6xy+y² using the Perfect Square Trinomial Identity.

Final Factorisation
The factorised form of 9x²+6xy+y² is (3x + y)². This means that the original expression can be expressed as the square of the sum of 3x and y.