Write the general equation of a parabola with its axis parallel to the x-axis.

The general equation for a parabola with its axis parallel to the x-axis is given by:

$$x = ay^2 + by + c$$

where $a$, $b$, and $c$ are constants to be determined.

Substituting $x = -1$ and $y = 2$:

$$-1 = a(2)^2 + b(2) + c$$ $$-1 = 4a + 2b + c \quad (*1)$$

Substituting $x = 1$ and $y = -1$:

$$1 = a(-1)^2 + b(-1) + c$$ $$1 = a - b + c \quad (*2)$$

Substituting $x = 2$ and $y = 1$:

$$2 = a(1)^2 + b(1) + c$$ $$2 = a + b + c \quad (*3)$$

Subtract equation $(*2)$ from equation $(*3)$:

$$(a + b + c) - (a - b + c) = 2 - 1$$ $$2b = 1$$ $$b = \frac{1}{2}$$

Substitute $b = \frac{1}{2}$ into equation $(*3)$:

$$a + \frac{1}{2} + c = 2$$ $$a + c = 2 - \frac{1}{2}$$ $$a + c = \frac{3}{2} \quad (*4)$$

Substitute $b = \frac{1}{2}$ into equation $(*1)$:

$$4a + 2\left(\frac{1}{2}\right) + c = -1$$ $$4a + 1 + c = -1$$ $$4a + c = -2 \quad (*5)$$

Subtract equation $(*4)$ from equation $(*5)$:

$$(4a + c) - (a + c) = -2 - \frac{3}{2}$$ $$3a = -\frac{4}{2} - \frac{3}{2}$$ $$3a = -\frac{7}{2}$$ $$a = -\frac{7}{6}$$

Substitute $a = -\frac{7}{6}$ into equation $(*4)$:

$$-\frac{7}{6} + c = \frac{3}{2}$$ $$c = \frac{3}{2} + \frac{7}{6}$$ $$c = \frac{9}{6} + \frac{7}{6}$$ $$c = \frac{16}{6}$$ $$c = \frac{8}{3}$$

Substitute the values of $a = -\frac{7}{6}$, $b = \frac{1}{2}$, and $c = \frac{8}{3}$ into the general equation $x = ay^2 + by + c$:

$$x = -\frac{7}{6}y^2 + \frac{1}{2}y + \frac{8}{3}$$

To eliminate fractions, multiply the entire equation by 6:

$$6x = -7y^2 + 3y + 16$$

Rearranging the terms to one side:

$$7y^2 - 3y + 6x - 16 = 0$$