Span

Determining the Span
Construct a linear equation by multiplying the spanning vectors V_1 and V_2 with arbitrary scalars, say x and y, and setting this equal to the given vector V.

x*(1,-1,2)+y*(2,1,3)=(3,3,4)

This gives you 3 simultaneous linear equations with 2 unknowns.

x*1+y*2=3 (1)
x*-1+y*1=3 (2)
x*2+y*3=4 (3)

Solve (1) and (2), obtain x and y, if the solution to another pair of equations gives the same x and y, then there is a unique solution to the system, which means those x and y values can be used to obtain V by a linear combination of V_1 and V_2, hence V lies in the span [V_1,V_2]

If solutions of x and y do not match, then there is no solution to the system indicating V does not lie in the span.

If you cannot obtain specific x and y, then there are infinitely many solutions and V does not lie in the span of V_1 and V_2.

This was a general answer, your case has a solution x=-1 and y=2. You can obtain vector V by -1*V_1+2*V_2 hence V lies in span [V_1,V_2]