“Let u = 1/x”.

Will look at this by 4 am if no one else has responded.

From a),

2x³ - 9x² + 3x + 4 = (2x + 1) (x - 1) (x - 4)


b)


2 + 3x² = 9x - 4x³

Divide both sides by x³,

2/x³ + 3x = 9/x² - 4

2/x³ - 9/x² + 3/x + 4 = 0

2(1/x)³ - 9(1/x²) + 3(1/x) + 4 = 0


Notice this is very similar to above. Effectively we are just replacing x with 1/x .


So if 2x³ - 9x² + 3x + 4 = (2x + 1) (x - 1) (x - 4),

2(1/x)³ - 9(1/x²) + 3(1/x) + 4 = (2/x + 1) (1/x - 1) (1/x - 4)

Then (2/x + 1) (1/x - 1) (1/x - 4) = 0

2/x = -1 or 1/x = 1 or 1/x = 4

x = -2 or x = 1 or x = 1/4

The key idea here is to notice that “2”, “-9”, “3” and “4” are present in both parts, but with the roles reversed.

Substitutions work very well here.

The idea is that the expressions look very similar (identical coefficients) and we should try to manipulate the -4x³ term in b) to get back to the constant term 4 in a). The rest will fall into place