Show that $x^4-8x^2+(1-m)x+1-c=(x-a)^2(x-b)^2$

Question

I am stuck on the following question:

My thoughts on how to approach it:

1. Find coefficient of $x$ in $(x-a)^2(x-b)^2$.
2. Equate coefficient to $1-m$
3. Prove that the values are equal. To prove it I thought of setting up $2$ simultaneous equations and finding $a, b, m$ - one where I write $f^\prime(a) = f^\prime(b) = m$ and other where I use $m = \frac{f(b) - f(a)}{b-a}$.
4. Use values of $a, b, m$ to find $c$ and prove that $1 - c = \text{coefficient of }x^0$

Problems with this approach:

1. It's a boring algebra bash that is too long and involves $4^{th}$ degree polynomials.
2. It doesn't follow the order of the questions i.e. $a, b, m$ are found before we prove part (c).
3. I am not sure whether it even works.

Any help on a simpler approach will be much appreciated.

Answer 1

Hints

$f(x)-g(x)=0$ has repeated roots at $x=a$ and $x=b$, then comparing coefficients (don't miss the $x^3$ term).

See also another post here.