Convert two polynomials into a target polynomials through operations 
You are provided with two polynomials $x^3-3x^2+2$ and $x^2-4x$ on a
piece of paper.
If there are polynomials $f(x)$ and $g(x)$ on the paper, you are allowed
to write $f(x) + g(x)$, $f(x) - g(x)$, $f(x)g(x)$, $f(g(x))$ and
$cf(x)$.
Can you obtain a linear polynomial (exactly one degree) given
two starting polynomials?

I feel like this could be a linear algebra problem, if the operations are just $f(x) + g(x)$, $f(x) - g(x)$ and $cf(x)$ Then we are looking for the linear dependence. But now sure how to handle $f(x)g(x)$, $f(g(x))$
 A: As mentioned, subtraction decreases the degree if the leading coefficients are the same (which we can ensure with $cf(x)$), so all we need is polynomials of the same degree. This is the crux. See if you can work it out before reading on.

We currently do not have 2 polynomials of the same degree, so let's force them out and take their difference. I will work with the specific cases of $a(x) = x^3 - 3x^2 + 2$, $b(x) = x^2 - 4x$, but I suspect this will generalize (with certain conditions).
Step 1: Calculate $a(x)a(x) = x^6 - 6x^5 + 9x^4 + 4x^3 - 12x^2 + 4$.
Step 2: Calculate $b(x)b(x)b(x) = x^6 - 12x^4 + 48x^2 - 64  $. This gives us 2 polynomials of degree 6.
Step 3: Take their difference $a(x)^2 - b(x)^3 = 6x^5 -39 x^4 + 68x^3 - 12x^2 + 4 = c(x) $.
Step 4: Calculate $a(x) b(x) = x^5 - 7x^4 + 12x^3 + 2x^2 - 8x $. This gives us 2 polynomials of degree 5.
Step 5: Take their difference $ c(x) - 6a(x) b(x) = -15x^4 + 86x^3 - 96x^2 + 48x + 4 = d(x)$.
Step 6: Calculate $b(x) b(x)  = x^4 - 8x^3 + 16x^2$. This gives us 2 polynomials of degree 4.
Step 7: Take their difference $d(x) + 15 6 b(x)b(x) = -34x^3 + 144x^2 + 48x + 4 = e(x) $, which is a polynomial of degree 3.
Step 8: Take their difference $ e(x) + 34 a(x) = 42x^2 + 48x + 72 = f(x) $, which is a polynomial of degree (at most, and hopefully) 2.
Step 9: Take their difference $ f(x)  - 42 b(x) = 216x+72 $, which is a polynomial of degree 1.

Notes

*

*As SenZen mentions, if our initial polynomials don't have a linear term, then none of the operations would have led to a linear term. So this isn't always doable.

*As implied by SenZen, we might lower the degree by more than one if the ratio of the first few coefficients are equal. This has happened when I was using $ a(b(x)) , b(a(x))$ as my starting polynomials, and I didn't end up with a polynomial of degree 1.

*This process is not unique. We could have made different choices for the polynomials of higher degrees.

*I might have a calculation error. If so, let me know. This was way to much calculation for a contest-math problem, even after knowing the crux, so I'd like to know of an alternative solution.

