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ANSWER
[tex]x \: > \: 17 [/tex]

EXPLANATION

The given function is

[tex]f(x) = 2 {x}^{2} - 31x - 51[/tex]
Let us split the middle term to get,

[tex]f(x) = 2 {x}^{2} -34x + 3x- 51[/tex]

We factor to obtain,

[tex]f(x) = 2x(x - 17) -+3(x - 17)[/tex]

[tex]f(x) = (2x -+3)(x -17) [/tex]

Let us find the zeros of f(x),

[tex] (2x +3)(x - 17) = 0[/tex]

Either

[tex]2x +3 = 0 \: or \: x - 17 = 0[/tex]
[tex]x = -\frac{3}{2} \: or \: x = 17[/tex]

We need to find the x-value of the vertex, which is the midpoint of the x-intercepts.

This gives us,

[tex] \frac{- \frac{3}{2} + 17}{2} = 7.75[/tex]
Hence the axis of symmetry of the parabola has equation,

[tex]x = 7.75[/tex]

Since the given function has minimum turning point, it will decrease and then increase.

The function will be increasing for
[tex]x \: > \: 7.75[/tex]

The function will be increasing and positive for
[tex]x \: > \: 17[/tex]

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