Linear Algebra Assistance: Linear Combinations of Vectors

a.) $\vec{v} = \left(\frac{-211}{2}\right)\vec{v_1}+42\vec{v_2}+ \left(\frac{279}{6}\right)\vec{v_3}$

To arrive at this answer, you can rewrite as a series of linear equations.

$a \vec{v_1} + b \vec{v_2} + c \vec{v_3} = \vec{v}$

You can write them in matrix form (being careful to rotate the coefficient so that each column represents the respective values of $\vec{v_1}$, $\vec{v_2}$, or $\vec{v_3}$) . Now that you have the equations, you can solve using any number methods. I did it by hand, but you could also  solve in Wolfram Alpha:

https://www.wolframalpha.com/input?i=LinearSolve%5B%5B%2F%2Fmath%3A%7B%7B2%2C1%2C4%7D%2C+%7B-1%2C1%2C-3%7D%2C+%7B2%2C5%2C0%7D%7D%2F%2F%5D%2C+%5B%2F%2Fmath%3A%7B17%2C8%2C-1%7D%2F%2F%5D%5D

You can check that this solution is correct using Wolfram Alpha: https://www.wolframalpha.com/input?i=%28-211%2F2%29%282%2C-1%2C2%29+%2B+42%281%2C1%2C5%29+%2B+%28279%2F6%29%284%2C-3%2C0%29

b.) $\vec{v} = \vec{v_1}+\vec{v_2}$
same method as above. Can solve using your favorite method:
https://www.wolframalpha.com/input?i=LinearSolve%5B%5B%2F%2Fmath%3A%7B%7B1%2C1%2C0%7D%2C+%7B1%2C0%2C1%7D%2C+%7B0%2C1%2C1%7D%7D%2F%2F%5D%2C+%5B%2F%2Fmath%3A%7B2%2C1%2C1%7D%2F%2F%5D%5D

You can check this by plugging in:

$ \vec{v_1}+\vec{v_2} = (1+ x) + (1+x^2) = 2 + x  + x^2$

BONUS: you can calculate the determinant of a matrix, and if it's zero then it's not solvable. In this case, both were solvable and had a non-zero determinant.

PS I assumed that the red dot above the $x^2$ term in $\vec{v_2}$ was just some random blip and had no meaning.