<s>[INST] If $f (x) +f'(x) = x^3+5x^2+x+2$ then find $f (x)$ 
If  $f (x) +f'(x) = x^3+5x^2+x+2$  then  find $f(x)$.

$f'(x)$ is the first derivative of $f (x)$.
I have no idea about this question, please help me. 
 [/INST] We work with the general form of the equation $f(x)$, assuming it to be a cubic equation: $$f(x)=ax^3+bx^2+cx+d$$
We differentiate this to see that $$f'(x)=3ax^2+2bx+c$$
Now we can say that \begin{align}f(x)+f'(x)&=(ax^3+bx^2+cx+d)+(3ax^2+2bx+c)\\
x^3+5x^2+x+2&=ax^3+(b+3a)x^2+(c+2b)x+(c+d)\end{align}
We can compare coefficients to solve this equation:
\begin{align}a&=1,\tag{$x^3$ term}\\\\
5&=b+3a\tag{$x^2$ term}\\
5&=b+3\times 1\\
&\Downarrow\\
b&=2,\\\\
1&=c+2b\tag{$x$ term}\\
1&=c+2\times 2\\
&\Downarrow\\
c&=-3,\\\\
2&=c+d\tag{constant term}\\
2&=(-3)+d\\
&\Downarrow\\
d&=5\end{align}
Therefore, we have that $$f(x)=x^3+2x^2-3x+5$$

We can check that this is a solution to the equation, by first computing $f'(x)$: 
$$f'(x)=3x^2+4x-3$$
and then checking the value of $f(x)+f'(x)$:
\begin{align}f(x)+f'(x)&=(x^3+2x^2-3x+5)+(3x^2+4x-3)\\
&=x^3+(2+3)x^2+(-3+4)x+(5+(-3))\\
&=x^3+5x^2+x+2\end{align}
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