In each part, determine whether the given vector p(1) in P₂ belongs to span (p₁(t), p₂(t), p3(1)), where p_{1}(t) = t ^ 2 + 2t + 1 p_{2}(t) = t ^ 2 + 3 and p_{1}(t) = t - 1 (a) p(t) = t ^ 2 + t + 2 (b) p(t) = 2t ^ 2 + 2t + 3 (c) p(t) = - t ^ 2 + t - 4 (d) p(t) = - 2t ^ 2 + 3t + 1

To determine whether a vector belongs to the span of a set of vectors, we need to check if it can be written as a linear combination of the vectors in the set. This means we need to find scalars a, b, and c such that:

p(t) = ap₁(t) + bp₂(t) + c*p₃(t)

Let's do this for each part:

(a) p(t) = t^2 + t + 2

We need to solve the system of equations:

a*(t^2 + 2t + 1) + b*(t^2 + 3) + c*(t - 1) = t^2 + t + 2

This system has no solution, so p(t) does not belong to the span of p₁(t), p₂(t), and p₃(t).

(b) p(t) = 2t^2 + 2t + 3

We need to solve the system of equations:

a*(t^2 + 2t + 1) + b*(t^2 + 3) + c*(t - 1) = 2t^2 + 2t + 3

This system has no solution, so p(t) does not belong to the span of p₁(t), p₂(t), and p₃(t).

(c) p(t) = -t^2 + t - 4

We need to solve the system of equations:

a*(t^2 + 2t + 1) + b*(t^2 + 3) + c*(t - 1) = -t^2 + t - 4

This system has no solution, so p(t) does not belong to the span of p₁(t), p₂(t), and p₃(t).

(d) p(t) = -2t^2 + 3t + 1

We need to solve the system of equations:

a*(t^2 + 2t + 1) + b*(t^2 + 3) + c*(t - 1) = -2t^2 + 3t + 1

This system has no solution, so p(t) does not belong to the span of p₁(t), p₂(t), and p₃(t).

In conclusion, none of the given vectors p(t) belong to the span of p₁(t), p₂(t), and p₃(t).