Answer:
Step-by-step explanation:
[tex]\frac{x^{2}+mx+n }{9-x^{2} } /\frac{x^{2} +6x-16}{x^{2} +px+q} = \frac{-x+3}{x+8}[/tex]
Change the division of the two fractions in to multiplication
[tex]\frac{x^{2}+mx+n }{9-x^{2} } * \frac{x^{2} +px+q}{x^{2} +6x-16} = \frac{-x+3}{x+8}[/tex]
Factor what we can
[tex]\frac{x^{2}+mx+n }{(3-x)(3+x)} * \frac{x^{2} +px+q}{(x-2)(x+8)} = \frac{(3-x)}{x+8}[/tex]
Cross multiply and simplify (x+8)/(x+8)
(x² +mx +n) (x² +px +q) = (x -3) (x -3) (x +3)( x -2)
Now we look to see what values for m, n, p, and q will give those factors.
m= 1 n= 2 p= 3 q= 4
(x² +x +2) (x² +3x +4) ≠ (x -3) (x -3) (x +3)( x -2) ❌
m= 1 n= -6 p= -6 q= 9
(x² +x -6) (x² -6x +9) = (x+3)(x-2) (x-3)² ✅
m= -5 n= 6 p= 0 q= -9
(x² -5x +6) (x² -9) = (x-3)(x-2)(x-3)(x+3)✅
m= 7 n= -4 p= 2 q= 0
(x² +7x -4) (x² +2x) ≠ (x-3)(x-2)(x-3)(x+3)❌
m= 3 n= -6 p= 1 q= 7
(x² +3x -6) (x² +x+7) ≠(x-3)(x-2)(x-3)(x+3)❌
m= -6 n= 9 p= 1 q= -6
(x² -6x +9) (x² +x -6) = (x-3)(x-3)(x-2)(x+3)✅
m= 0 n= -9 p= -5 q= 6
(x² -9) (x² -5x +6) ≠(x-3)(x-3)(x+3)(x-2)❌
