Given $W=\{f(t) \in \mathbb{R}[x] \mid \deg f \leqslant 2,\ f(1)=0\}$, find the dimension of $W$ and a basis for $W$.
$$f(t)=a+bt+ct^2,\quad f(1)=a+b+c=0.$$ I think $\dim W=3$ and I use $a+b+c=0$ to form the basis by put the number inside the equation and make it become $0$.
From $a+b+c=0$, take for example $a=-b-c$ so an arbitray $f(t) \in W$ will be of the form: $$f(t)=-b-c+bt+ct^2=b\left(-1+t\right)+c\left(-1+t^2\right)$$ This means you can write any $f(t) \in W$ as a linear combination of $-1+t$ and $-1+t^2$.
These two are linearly independent, so...?