I think it's by definition, for a function to be continuous

lim f(x) as x --> a = f(a)

which also means

lim f(x) as x --> a+ = f(a) = lim f(x) as x --> a-

where lim means "limit as", x --> a+ means keeping x > a but taking it towards a and x ---> a- means x approaches a from its lower side.

So...

at x = 1

lim (x+1) as x-->a+ = lim(x^2+bx+c) as x-->a- where a = 1

1 + 1 = 1 + b + c or b + c = 1

Do the same for x --> 3+ and 3-. Find b and c.