How to square both the sides of an equation? Question: $x^2  \sqrt{(x + 3)} = (x + 3)^{3/2}$
My solution: $x^4 (x + 3) = (x + 3)^3$
$=> (x + 3)^2 = x^4$
$=> (x + 3)  = x^2$
$=> x^2 -x - 3 = 0$
$=> x = (1 \pm \sqrt{1 + 12})/2$
I understand that you can't really square on both the sides like I did in the first step, however, if this is not the way to do it, then how can you really solve an equation like this one (in which there's a square root on the LHS) without substitution? 

You can square both sides of the equation the way you did. But there is a problem in the third line of your working.
You had $x^{4}(x+3)=(x+3)^{3}$ and then divided both sides by $(x+3)$ to get $x^{4}=(x+3)^{2}$.
This is a problem because you are losing solutions to the overall equation.
It is better to use factorisation to solve the equation for $x$.
