Find the values of $$ a, b, c $$ and $$ d $$ for which $$ \frac{9 x^{4}+12 x^{3}+3 x^{2}+5 x-1}{3 x^{3}+x}=a x+b+\frac{c x+d}{3 x^{3}+x} $$

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We start with the equation

$$
\frac{9x^4+12x^3+3x^2+5x-1}{3x^3+x} = ax + b + \frac{cx+d}{3x^3+x}.
$$

Multiply both sides by $$3x^3+x$$ to obtain

$$
9x^4+12x^3+3x^2+5x-1 = (3x^3+x)(ax+b) + (cx+d).
$$

Next, we expand the product $$ (3x^3+x)(ax+b) $$:

$$
(3x^3+x)(ax+b) = 3a x^4 + 3b x^3 + a x^2 + b x.
$$

So the equation becomes

$$
9x^4+12x^3+3x^2+5x-1 = 3a x^4 + 3b x^3 + a x^2 + b x + cx + d.
$$

Now, we group like terms on the right-hand side:

- Coefficient of $$x^4$$: $$3a$$
- Coefficient of $$x^3$$: $$3b$$
- Coefficient of $$x^2$$: $$a$$
- Coefficient of $$x$$: $$b+c$$
- Constant term: $$d$$

Set up the equations by equating the coefficients with the left-hand side:

1. For $$x^4$$: 
   $$
   3a = 9 \quad \Longrightarrow \quad a = 3.
   $$
2. For $$x^3$$:
   $$
   3b = 12 \quad \Longrightarrow \quad b = 4.
   $$
3. For $$x^2$$:
   $$
   a = 3 \quad \text{(which is consistent with $$a=3$$)}.
   $$
4. For $$x$$:
   $$
   b + c = 5 \quad \Longrightarrow \quad c = 5 - 4 = 1.
   $$
5. For the constant term:
   $$
   d = -1.
   $$

Thus, the values are

$$
a = 3,\quad b = 4,\quad c = 1,\quad d = -1.
$$
$$ a = 3 $$, $$ b = 4 $$, $$ c = 1 $$, $$ d = -1 $$.

Find the values of and for which

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