To solve the expression $a(-a) + 1$:

1. First, simplify the product $a(-a)$: $a \cdot (-a) = -a^2$

2. Next, add 1 to the result: $-a^2 + 1$

So, the simplified expression is: $-a^2 + 1$

Would you like any further details or have any questions about this solution?

Here are some related questions to further explore this topic:

1. How do you solve for $a$ if given an equation like $a(-a) + 1 = 0$?
2. What are the properties of exponents when multiplying variables?
3. How do you handle negative coefficients in algebraic expressions?
4. Can you graph the function $f(a) = -a^2 + 1$ and describe its properties?
5. What is the vertex of the parabola $y = -a^2 + 1$?
6. How does the sign of the coefficient in front of $a^2$ affect the direction of the parabola?
7. What is the axis of symmetry for the parabola $y = -a^2 + 1$?
8. How do you determine the maximum value of the quadratic function $y = -a^2 + 1$?

Tip: When simplifying algebraic expressions, always handle the operations step-by-step, keeping track of positive and negative signs carefully.