Answer
linear dependence relation: $1 \cdot \mathbf{p}_1 + 1 \cdot \mathbf{p}_2 - 2 \cdot \mathbf{p}_3 = 0$
basis: $\{ \mathbf{p}_1, \mathbf{p}_2 \}$
Work Step by Step
That the given linear combination evaluates to $\mathbf{0}$ is easy to verify. To see that $\{ \mathbf{p}_1, \mathbf{p}_2 \}$ constitutes a basis for $\text{Span} \{\mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3 \}$, we need only see that the linear dependence relationship we found may be rewritten $$\frac{1}{2} \cdot \mathbf{p}_1 + \frac{1}{2} \cdot \mathbf{p}_2 = \mathbf{p}_3.$$ Hence, $\mathbf{p}_3$ is a linear combination of the other elements, so we can remove it from the set $\{ \mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3 \}$ without affecting the span. But $\mathbf{p}_1$ and $\mathbf{p}_2$ are linearly independent, since neither is a scalar multiple of the other, so they constitute a basis for the span.
In fact, it is not difficult to see that any two of the three polynomials form a linearly independent set, so any two of $\mathbf{p}_1$, $\mathbf{p}_2$, $\mathbf{p}_3$ forms a basis for $\text{Span} \{\mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3 \}$.
