I'm new to straight line equation and trying to find out whether $y (2 + x) = 3$ represents a straight line equation or not. Could anybody please help me to figure out how to reach a conclusion here.
No it does not, It represents a hyperbola.
General equation of a straight line is $ax+by+c=$ where $a,b,c \in\mathbb R$. But your equation when simplified looks like $2y+xy-3=0$ Notice that straight line equation has no $xy$( or coefficient term $xy$ term is zero) hence it does not represent a straight line.
There's a simple way to distinguish between a linear equation (or any polynomial) and a non-polynomial equation. A polynomial is defines as $$y=a_1x^1 + a_2x^2 + ... + a_nx^n$$ Observe the variable $x$ is on one side and $y$ is on the other, this is the general form. So bring one of the variable to one side and see if the degree is a positive integer, only then the equation is a polynomial.
Try using this information for your question. (A linear equation is an equation of degree 1)