Answer
$273$
Work Step by Step
Let's note:
$$S=\sum_{i=1}^6 3i^2.$$
Factor out $3$:
$$S=3\sum_{i=1}^6. i^2$$
We know the formula for the sum of squares:
$$\sum_{i=1}^n i^2=\dfrac{n(n+1)(2n+1)}{6}\tag1$$
Use Eq. $(1)$ for $n=6$ to calculate our sum:
$$S=3\cdot\dfrac{6(6+1)(2(6)+1)}{6}=273.$$
