When the graph of $y = 2x^2 - x + 7$ is shifted four units to the right, we obtain the graph of $y = ax^2 + bx + c$.  Find $a + b + c$.
To shift the graph of $y = 2x^2 - x + 7$ four units to the right, I need to replace $x$ with $x - 4$ in the equation. So I get $y = 2(x - 4)^2 - (x - 4) + 7$. I need to simplify this expression and write it in the form $y = ax^2 + bx + c$. I can start by expanding the squared term using the binomial formula: $y = 2(x^2 - 8x + 16) - x + 4 + 7$. Then I can distribute the 2 and combine like terms: $y = 2x^2 - 16x + 32 - x + 11$. Finally, I can write it as $y = 2x^2 - 17x + 43$. Now I can see that $a = 2$, $b = -17$, and $c = 43$. To find $a + b + c$, I just need to add these coefficients: $a + b + c = 2 - 17 + 43$. 