First you need to convert the information given in the form of range to heaviside:
f(t) = 0, 0 < t < 2;
→ 0.[H(t-0) - H(t-2)]
f(t) = (t+1), t > 2;
→ (t+1). H(t+2)
(t-2)H(t-2) + 3H(t-2)
then you apply laplace transform to each of the terms above:
for the first term you will get: 0
for the second term you will get(using table):
exp(-2s)/(s^2) +3 exp(-2s)/s
for the second part just change (t+1) to 1 and repeat the steps again.
and yeah you can always use the general method posted by TheEmptySet.