We have to prove the identity 3+(4+x)^3=x^3+12x^2+115

Starting with the left hand side

3+(4+x)^3

=> 3 + 4^3 + x^3 + 3*16*x + 3*4*x^2

=> 3 + 64 + x^3 + 48x + 12x^2

=> x^3 + 12x^2 + 48x + 67

We see that this is not equal to the right hand side.

**Therefore the expression that we have is not an identity that can be proved.**

We'll manage the left side and we'll expand the cube:

(4+x)^3 = 4^3 + x^3 + 3*4*x(4+x)

(4+x)^3 = 64 + x^3 + 12*x(4+x)

We'll remove the brackets from the right:

(4+x)^3 = 64 + x^3 + 12*x^2 + 48x

We'll add 3 both sides:

(4+x)^3 + 3 = 64 + x^3 + 12*x^2 + 48x + 3

We'll combine like terms:

(4+x)^3 + 3 = x^3 + 12*x^2 + 48x + 67

We notice that the right side of the identity to be demonstrated does not have the term in x, so, the both sides expressions are not equivalent.

**The given expression does not represent an identity.**