Explain why this question is a quadratic.

The question is a quadratic because the mini 2 [ which means squared ] is there and as simple as that the question becomes a quadratic :D.

Y = 3(x + 1)²
y = 3(x² + 2x + 1)
y = 2(x² +2x + 1)
y = 2x² + 4x + 2

Here, You can compare it with principle Quadratic equation, 
ax² + bx + c = 0
So, as the pattern matches, it is a "Quadratic Equation"

Hope this helps!