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The given problem asks to find the Lagrange interpolation polynomial for the points:

\[
(1, -3), (3, 0), (4, 30), (6, 132)
\]

and then compute \( y(5) \).

### Step 1: Lagrange Interpolation Formula

The Lagrange interpolation polynomial \( P(x) \) is given by:

\[
P(x) = \sum_{i=1}^{n} y(x_i) L_i(x)
\]

where \( L_i(x) \) are the Lagrange basis polynomials, defined as:

\[
L_i(x) = \prod_{\substack{1 \leq j \leq n \\ j \neq i}} \frac{x - x_j}{x_i - x_j}
\]

For this case, we have the points \( x_1 = 1, x_2 = 3, x_3 = 4, x_4 = 6 \) with corresponding \( y \)-values: \( y(1) = -3, y(3) = 0, y(4) = 30, y(6) = 132 \).

Let me compute the polynomial and find \( y(5) \).

Find Lagrange's interpolation polynomial fitting the points y(1) = -3, y(3) = 0, y(4) = 30, y(6) = 132. Hence find y(5).

This problem involves finding the Lagrange interpolation polynomial for the points y(1) = -3, y(3) = 0, y(4) = 30, and y(6) = 132. The solution demonstrates step-by-step how to compute y(5) using polynomial interpolation methods, making it suitable for undergraduate students.

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