To evaluate the given expression $-2(x - 4)^2 + 11$ when $x = 1$, we will substitute $x = 1$ into the expression and simplify.

Here are the steps:

1. Substitute $x = 1$ into the expression:

   $-2(1 - 4)^2 + 11$

2. Simplify the expression inside the parentheses:

   $1 - 4 = -3$

   So the expression now is:

   $-2(-3)^2 + 11$

3. Calculate $(-3)^2$:

   $(-3)^2 = 9$

   So the expression now is:

   $-2 \cdot 9 + 11$

4. Multiply $-2$ by $9$:

   $-2 \cdot 9 = -18$

   So the expression now is:

   $-18 + 11$

5. Add $-18$ and $11$:

   $-18 + 11 = -7$

Thus, the value of the expression $-2(x - 4)^2 + 11$ when $x = 1$ is $-7$.

Would you like more details or have any questions?

Here are five related questions to consider:

1. How would the expression change if $x$ were a different value?
2. What is the general form of the expression after expanding $-2(x - 4)^2 + 11$?
3. How does the graph of $y = -2(x - 4)^2 + 11$ look, and what are its vertex and axis of symmetry?
4. How would you solve for $x$ if the expression equals zero?
5. How does changing the coefficient $-2$ affect the shape of the parabola?

Tip: When working with expressions involving squares, always remember to handle negative signs carefully, especially when squaring negative numbers, as $(-a)^2$ is positive.