Answer
False
Work Step by Step
Let $S=\{1+x^2,x+x^2,x^2\}$ be the vector space of $3 × 3$ upper triangular matrices.
Putting a polynominal $p(x)=a+bx+cx^2$ into set $S$ we have:
$a(1+x^2)+b(x+x^2)+(-a-b+c)x^2=a+ax^2+bx+bx^2-ax^2-bx^2+cx^2=a+bx+cx^2=p(x)$
Hence, we can say $S$ span $P_2$
The statement is true.