The Laplace function is:
ℒ{f(t)}=∫[0-,∞]e-stf(t)dt, so to verify a transform, you need to evaluate the integral for f(t). Note that s is considered to be constant for the purposes of integration wrt t, so we end up with F(s)=ℒ{f(t)}.
You can use a table of transforms to avoid the labour of doing the calculations yourself. Such a table will also list inverse transforms, which would otherwise be difficult to work out using calculus. When attempting to solve a problem requiring Laplace Transforms, consult a table first to find out what transforms are listed, then examine the problem to make best use of the transforms. This may require reducing the problem to partial fractions, for example, or some other type of simplification that indicates which transforms to use. Remember that constants and add/subtract operations are transparent to the transformation process. That is, ℒ{kf(t)}=kℒ{f(t)}, and ℒ{af(t)+bg(t)-ch(t)}=aℒ{f(t)}+bℒ{g(t)}-cℒ{h(t)}. You cannot use multiplication or division of transforms, that is, ℒ{f(t)g(t)/h(t)}≠ℒ{f(t)×ℒ{g(t)}/ℒ{h(t)}. The expression {f(t)g(t)/h(t) would need to be reduced to an expression involving only addition or subtraction and constant coefficients of the functions.
Not all functions have a Laplace Transform. The definition of the transform involves integration and you cannot integrate a non-integrable function. An example would be a function which is discontinuous or is undefined in the domain [0-,∞]. 0- requires continuity for all t between slightly less than zero and infinity.