The problem provides a table of values for the relation $y = px^2 - 5x + q$. We need to find the values of $p$ and $q$, and then complete the table.

1. Problem Description

The problem provides a table of values for the relation $y = px^2 - 5x + q$. We need to find the values of $p$ and $q$, and then complete the table.

2. Solution Steps

(a) (i) Finding the values of $p$ and $q$:

We can use the given data points from the table to form a system of equations to solve for $p$ and $q$. We will use the points $(0, -12)$ and $(4, 0)$.

Using the point $(0, -12)$, we substitute $x=0$ and $y=-12$ into the equation $y = px^2 - 5x + q$:

$-12 = p(0)^2 - 5(0) + q$

$-12 = 0 - 0 + q$

$q = -12$

Now we have $y = px^2 - 5x - 12$.

Using the point $(4, 0)$, we substitute $x=4$ and $y=0$:

$0 = p(4)^2 - 5(4) - 12$

$0 = 16p - 20 - 12$

$0 = 16p - 32$

$16p = 32$

$p = \frac{32}{16}$

$p = 2$

Thus, $p = 2$ and $q = -12$. The equation is $y = 2x^2 - 5x - 12$.

(a) (ii) Completing the table:

Now that we have the equation $y = 2x^2 - 5x - 12$, we can complete the table by substituting the given values of $x$:

For $x = -1$:

$y = 2(-1)^2 - 5(-1) - 12 = 2(1) + 5 - 12 = 2 + 5 - 12 = 7 - 12 = -5$

For $x = -2$:

$y = 2(-2)^2 - 5(-2) - 12 = 2(4) + 10 - 12 = 8 + 10 - 12 = 18 - 12 = 6$ (Given)

For $x = -3$:

$y = 2(-3)^2 - 5(-3) - 12 = 2(9) + 15 - 12 = 18 + 15 - 12 = 33 - 12 = 21$ (Given)

For $x = 1$:

$y = 2(1)^2 - 5(1) - 12 = 2(1) - 5 - 12 = 2 - 5 - 12 = -3 - 12 = -15$

For $x = 2$:

$y = 2(2)^2 - 5(2) - 12 = 2(4) - 10 - 12 = 8 - 10 - 12 = -2 - 12 = -14$

For $x = 3$:

$y = 2(3)^2 - 5(3) - 12 = 2(9) - 15 - 12 = 18 - 15 - 12 = 3 - 12 = -9$

For $x = 5$:

$y = 2(5)^2 - 5(5) - 12 = 2(25) - 25 - 12 = 50 - 25 - 12 = 25 - 12 = 13$ (Given)

So, the completed table is:

$x$ | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5

------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | --------

$y$ | 21 | 6 | -5 | -12 | -15 | -14 | -9 | 0 | 13

3. Final Answer

$p=2$ and $q=-12$.

The completed table is:

$x$ | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5

------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | --------

$y$ | 21 | 6 | -5 | -12 | -15 | -14 | -9 | 0 | 13