Is 1/4 the correct answer? To the box before and how did you solve it

Question

Solve 7/x-4-5x+3=2/x^2-x-12

Answer 1

The equation 7/(x - 4) - 5x + 3 = 2/(x² - x - 12)

7/(x - 4) - 5x + 3 = 2/[(x  + 3)(x - 4)]

7/(x - 4) - 5x + 3 - 2/[(x  + 3)(x - 4)]  = 0

[ 7(x + 3) - 5x(x  + 3)(x - 4) + 3(x  + 3)(x - 4) - 2 ]/(x - 4)(x + 3) = 0

7x + 21 - (5x² + 15x)(x - 4) + (3x + 9)(x - 4) - 2  = (0)(x - 4)(x + 3)

 7x + 21 - (5x³ - 20x² + 15x² - 60x) + (3x² - 12x + 9x - 36 - 2) = 0

 7x + 21 - (5x³ - 5x² - 60x) + (3x² - 3x - 38) = 0

7x + 21 - 5x³ + 5x² + 60x + 3x² - 3x - 38 = 0

 - 5x³ + 8x² + 64x - 17 = 0

Solve the above cubic polynomial using factoring.

Graph

The points where it crosses the x axis  will give solutions to the polynomial function .

The graph crosses the x - axis at a point that would suggest a factor.

It crosses the x - axis at three points hence there are 3 real roots.

 Now x  = -3.0166

Use synthetic division to determine if the given value of x is a root of the polynomial.

Since f(-0.6503) = 0, x = -0.6503 is a zero.

The depressed polynomial is  - 5x² + 23.083x - 5.6321

Since the depressed polynomial of this zero, - 5x² + 23.083x - 5.6321 = 0, is quadratic,

Use the Quadratic Formula to find the roots of the related quadratic equation ax² + bx + c = 0

a = -5, b = 23.083, c = - 5.6321

Roots are

Solutions are x = 0.2585 , 4.3581 and -3.0661.

One of the solution is approximately x = 1/4.