What is the maximum possible dimension of a
proper subspace of the vector space of polynomials in \(x\)
with real coefficients having degree at most \(2\)?
The answer is 2.
The dimension of the vector space of polynomials in \(x\) having
degree at most two is \(3\).
Hence, any proper subspace can have at most dimension \(2\).
Dimension 2 is indeed possible. For example, the span of \(\{1, x\}\) is
a proper subspace having dimension 2.