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To ensure the function \( f(x) \) is continuous on the entire real line, particularly at the point \( x = -6 \), the left-hand limit of the function as \( x \) approaches \(-6\) from the left must equal the right-hand limit as \( x \) approaches \(-6\) from the right, which should also equal the function value at that point.

The function is defined as:

\[
f(x) = 
\begin{cases}
\frac{2x^3 + 13x^2 + 14x + 48}{x + 6} & \text{if } x < -6, \\
x^2 + 2x + a & \text{if } x \geq -6.
\end{cases}
\]

### Step 1: Simplify the first piece of the function.

To simplify the expression \(\frac{2x^3 + 13x^2 + 14x + 48}{x + 6}\), we perform polynomial division or factorization. 

Let's factorize the numerator to simplify it.

### Step 2: Find the limit from the left as \( x \to -6 \).

Simplify the expression and evaluate the limit as \( x \to -6 \).

### Step 3: Find the value of the function from the right as \( x \to -6 \).

Evaluate the expression \( x^2 + 2x + a \) at \( x = -6 \).

### Step 4: Set both limits equal and solve for \( a \).

Let's carry out these steps to find the required value of \( a \).Here's the detailed evaluation:

1. **Simplification of the left-hand expression**:
   - The simplified form of the function as \( x \to -6 \) gives us a limit of \( 74 \).

2. **Right-hand evaluation**:
   - The right-hand expression \( x^2 + 2x + a \) evaluated at \( x = -6 \) is \( a + 24 \).

3. **Equation to solve**:
   - Setting the left limit equal to the right value:
     \[
     74 = a + 24
     \]

4. **Solving for \( a \)**:
   - Solving this equation, we get:
     \[
     a = 50
     \]

Thus, the value of the constant \( a \) that makes the function continuous on \( (-\infty, \infty) \) is **50**.

Would you like more details on any step or have any questions?

### Related Questions:
1. How do you factorize a polynomial to simplify a rational expression?
2. What is the concept of limits in ensuring continuity?
3. How can you use polynomial division to simplify rational functions?
4. What is the role of continuity at a specific point in a piecewise function?
5. How does solving equations help in determining constants for continuity?

**Tip**: When working with continuity problems, always equate the left and right-hand limits at the point of interest for a piecewise function!

Find the value of the constant a that makes the following function continuous on (-∞, ∞). f(x) = { (2x^3 + 13x^2 + 14x + 48) / (x + 6), if x < -6; x^2 + 2x + a, if x >= -6 }

Learn how to find the value of the constant 'a' that makes a piecewise function continuous on the interval (-∞, ∞). This solution covers key concepts like polynomial division, limits, and the continuity condition at a specific point.

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