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The problem provided asks for the two integer values of \( a \) that make the given function continuous at every \( x \). The piecewise function is:

\[
g(x) = 
\begin{cases} 
a x^2 + 2a x & \text{if } x \leq 1 \\
3 & \text{if } x > 1
\end{cases}
\]

To ensure the function is continuous at \( x = 1 \), the two pieces of the function must be equal at \( x = 1 \). 

### Steps:

1. **Continuity Condition**: For the function to be continuous at \( x = 1 \), the limit of \( g(x) \) as \( x \) approaches 1 from the left must equal the limit as \( x \) approaches 1 from the right. Thus, we set the two pieces equal at \( x = 1 \):
   \[
   a(1)^2 + 2a(1) = 3
   \]
   
2. **Simplifying**:
   \[
   a + 2a = 3
   \]
   \[
   3a = 3
   \]
   \[
   a = 1
   \]

Thus, the value \( a = 1 \) ensures the function is continuous.

### Verification:
To confirm, let's plug \( a = 1 \) back into the function. For \( x \leq 1 \), the function becomes:
\[
g(x) = 1x^2 + 2(1)x = x^2 + 2x
\]
At \( x = 1 \):
\[
g(1) = 1^2 + 2(1) = 1 + 2 = 3
\]
This matches the value of the function for \( x > 1 \), which is also 3. Therefore, the function is continuous for \( a = 1 \).

### Conclusion:
The function is continuous for \( a = 1 \).

Would you like further details or any additional questions answered on this topic?

Here are five related questions for practice:
1. What does it mean for a function to be continuous at a point?
2. How do you verify if a function is continuous on an interval?
3. Can a function be continuous if it has different expressions on different intervals?
4. What role do limits play in determining continuity of a function?
5. How would the function change if it were discontinuous at \( x = 1 \)?

**Tip**: When solving for continuity, always check that the left-hand and right-hand limits match at the point of concern.

For what two integer values of a is the following function continuous at every x?
g(x) = {a * x^2 + 2a * x if x <= 1, 3 if x > 1}

Solve for integer values of a that make the given piecewise function continuous. This problem involves concepts of continuity, algebra, and piecewise functions, suitable for students in Grades 10-12.

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