10x^2-3x-3= A(x^2+1)+(Bx+C) (x-1)
find A B C
y=kx-5
y=6x-x^2-6
solve for k
$\displaystyle 10x^2-3x-3 =A(x^2+1)+(Bx+C)(x-1) $
$\displaystyle 10x^2-3x-3=Ax^2+A +Bx^2-Bx+xC-C$
the factors which multiply with $\displaystyle x^2$ in the right side should equal to the factor of $\displaystyle x^2$ in the left side so you will have
$\displaystyle 10=A+B$
and the factors which are multiply with x in the right side should equal to the factor with x in the left side so you will have
$\displaystyle -3=-B+C$
and finally the terms with no x in the right side should equal to the terms with no x in the left side so you will have this
$\displaystyle -3 = A -C$
now you have three equation with three variables so you can solve them
$\displaystyle 10=A+B$
$\displaystyle -3=-B+C$
$\displaystyle -3 = A -C$
find the sum of the second and the third
$\displaystyle -6=A-B$ find the sum of this and the first one
$\displaystyle 4=2A \Rightarrow A=2 $ the rest for you
Multiply out the right side: $\displaystyle 10x^2- 3x- 3= Ax^2+ A+ Bx^2- Bx+ Cx- C$.
$\displaystyle 10x^2- 3x- 3= (A+ B)x^2+ (C- B)x- C$
In order that these be equal for all x, corresponding coefficients must be equal: 10= A+ B, -3= C- B, and -3= -C. Solve for A, B, and C.
$\displaystyle y= kx- 5= 6x- x^2- 6$. Because the left side does not have an "$\displaystyle x^2$" term, there is NO value of k that will make this true for all x.y=kx-5
y=6x-x^2-6
solve for k