(c) $$ g^{2} \frac{2 x}{2} $$ Determine the numerical value of $$ (3 x+p)^{2} $$ if it is given that $$ 9 x^{3}+y^{2}=12 $$ and $$ x y=-3 $$ Given that $$ m=x(x-y)^{2} $$. Determine the value of $$ m $$ if $$ x y^{2}=4 $$ and $$ x^{3}-2 x^{2} y=3 $$

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Alright, I need to determine the numerical value of $$ (3x + p)^2 $$ given the equations $$ 9x^3 + y^2 = 12 $$ and $$ xy = -3 $$. Additionally, I need to find the value of $$ m = x(x - y)^2 $$ given $$ xy^2 = 4 $$ and $$ x^3 - 2x^2 y = 3 $$. Let's tackle these problems step by step.

**First Problem: Determining $$ (3x + p)^2 $$**

Given:
1. $$ 9x^3 + y^2 = 12 $$
2. $$ xy = -3 $$

I need to find $$ (3x + p)^2 $$. However, the value of $$ p $$ is not provided. This seems like a missing piece of information. Without knowing the value of $$ p $$, I can't compute a numerical value for $$ (3x + p)^2 $$. Maybe there's an assumption I can make about $$ p $$, but it's not specified. Perhaps $$ p $$ is a constant or related to $$ x $$ and $$ y $$ in some way. Without additional information, I can't proceed further with this part.

**Second Problem: Determining $$ m = x(x - y)^2 $$**

Given:
1. $$ xy^2 = 4 $$
2. $$ x^3 - 2x^2 y = 3 $$

I need to find the value of $$ m = x(x - y)^2 $$. Let's see if I can find expressions for $$ x $$ and $$ y $$ from the given equations.

From the first equation:
$$ xy^2 = 4 $$
This can be rewritten as:
$$ y^2 = \frac{4}{x} $$
Taking the square root of both sides:
$$ y = \pm \frac{2}{\sqrt{x}} $$

From the second equation:
$$ x^3 - 2x^2 y = 3 $$
Substituting $$ y $$ from the first equation:
$$ x^3 - 2x^2 \left( \pm \frac{2}{\sqrt{x}} \right) = 3 $$
Simplifying:
$$ x^3 - \frac{4x^2}{\sqrt{x}} = 3 $$
$$ x^3 - 4x^{3/2} = 3 $$
This equation is a bit complex. Maybe I can make a substitution to simplify it. Let $$ t = x^{1/2} $$, so $$ x = t^2 $$. Substituting:
$$ (t^2)^3 - 4(t^2)^{3/2} = 3 $$
$$ t^6 - 4t^3 = 3 $$
Let $$ u = t^3 $$, then:
$$ u^2 - 4u - 3 = 0 $$
Solving this quadratic equation for $$ u $$:
$$ u = \frac{4 \pm \sqrt{16 + 12}}{2} $$
$$ u = \frac{4 \pm \sqrt{28}}{2} $$
$$ u = 2 \pm \sqrt{7} $$
Since $$ u = t^3 $$ and $$ t = x^{1/2} $$, we have:
$$ t^3 = 2 \pm \sqrt{7} $$
Taking the cube root:
$$ t = \sqrt[3]{2 \pm \sqrt{7}} $$
Then:
$$ x = t^2 = \left( \sqrt[3]{2 \pm \sqrt{7}} \right)^2 $$
This gives two possible values for $$ x $$, depending on the sign chosen for $$ \sqrt{7} $$. Once $$ x $$ is determined, $$ y $$ can be found using $$ y = \pm \frac{2}{\sqrt{x}} $$.

Now, to find $$ m = x(x - y)^2 $$, I'll substitute the expressions for $$ x $$ and $$ y $$:
$$ m = x \left( x - \left( \pm \frac{2}{\sqrt{x}} \right) \right)^2 $$
$$ m = x \left( x \mp \frac{2}{\sqrt{x}} \right)^2 $$
$$ m = x \left( x^2 \mp 4x + \frac{4}{x} \right) $$
$$ m = x^3 \mp 4x^2 + 4 $$
But from the second given equation $$ x^3 - 2x^2 y = 3 $$, I can express $$ x^3 $$ in terms of $$ x^2 $$ and $$ y $$:
$$ x^3 = 3 + 2x^2 y $$
Substituting back:
$$ m = (3 + 2x^2 y) \mp 4x^2 + 4 $$
$$ m = 3 + 2x^2 y \mp 4x^2 + 4 $$
$$ m = 7 + 2x^2 y \mp 4x^2 $$
This expression still contains $$ x^2 y $$, which can be expressed in terms of $$ x $$ using the first given equation $$ xy^2 = 4 $$. However, without specific values or additional relationships, it's challenging to simplify further. It seems that without knowing the exact values of $$ x $$ and $$ y $$, or another equation relating them, I can't compute a numerical value for $$ m $$.

**Conclusion:**
- For the first problem, the value of $$ (3x + p)^2 $$ cannot be determined without knowing the value of $$ p $$.
- For the second problem, while I can express $$ m $$ in terms of $$ x $$ and $$ y $$, without additional information or specific values for $$ x $$ and $$ y $$, a numerical value for $$ m $$ cannot be determined.

Cannot determine the numerical value of $$ (3x + p)^2 $$ without knowing the value of $$ p $$. For $$ m = x(x - y)^2 $$, the value cannot be determined with the given information.

(c) \( g^{2} \frac{2 x}{2} \) Determine the numerical value of \( (3 x+p)^{2} \) if it is given that \( 9 x^{3}+y^{2}=12 \) and \( x y=-3 \) Given that \( m=x(x-y)^{2} \). Determine the value of \( m \) if \( x y^{2}=4 \) and \( x^{3}-2 x^{2} y=3 \)

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