Question

(x-1)(x-2)(2x-1)(2x+1)-70=0

Answer 1

(x-1)(x-2(((2x-1)(2x+1))-70=0
here (2x-1)(2x+1) is in the form of (a+b)(a-b)
formulae for it is a^2-b^2
here a=2x and b=1
substituting the values we get
(2x-1)(2x+1)=(2x)^2-(1)^2
                    =4x^2-1 
(x-1)(x-2)=x(x-2)-1(x-2)
               =x^2-2x-x+2
               =x^2-x+2
(x-1)(x-2)(2x-1)(2x+1)-70=0
(x^2-x+2)(4x^2-1)-70=0
By simplification we get
4x^4-12x^3+7x^2+3x-72=0
By long division method we can write as
(x-3)(4x^3+7x+24)=0
Again by long division method we can write as
(x-3)(2x+3)(2x^2-3x+8)=
by using the principle of zero products
x-3=0 or 2x+3=0 or 2x^2-3x+8=0
Thus the values are
x-3=0
x=3 
2x+3=0
2x=-3
x=-3/2
2x^2+3x+2=0
By simplification
x=0.7500+1.8540i and x=0.7500-1.8540i

Answer 2

$(x-1)(x-2)(2x-1)(2x+1)-70=0\\ \\(x^2-3x+2)(4x^2-1)-70=0\\ \\4x^4-x^2-12x^3+3x+8x^2-2-70=0\\ \\4x^4-12x^3+7x^2+3x-72=0$

The rational factors, if they exist, are one or some of the following :
          +-  factors of 72 / factors of 4
      =  +- 1, 2,3,4, 6,8,9,12,18,36,72,   +- 1/2, 1/4, 3/2, 3/4, 9/2,  9/4 

On checking them you find that  x = 3 and x = -3/2  satisfy the polynomial.

Let us rewrite the polynomial in the following way. and find the value of "a"

     (x-3) (x+3/2) 4 (x² + a x - 72/(4*-3*3/2) ) = 0
       (x² - 3/2 x - 9/2) 4 (x² + a x + 4) = 0
Coefficient of x³ will be : 4 a - 4 * 3/2 = - 12
                             a = -3/2

So   x² - 3/2 x + 4 = 0

$x = \frac{1.5 +- \sqrt{2.25-16}}{2}=0.75+ 1.854\ i , \ \ \ 0.75-1.854\ i$

So there are two real and two imaginary roots.

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Trying the rational roots:

$P(x)=4x^4-12x^3+7x^2+3x-72=0$

P(0) = -ve   P(4) = 308 = +ve.  so a root exists between x= 0 & 4.
      Trying   P(3) =0   x = 3 is a root.

P(-4) = 1820 = +ve   So a root exists between 0 and -4.
          Trying  -1, -2, and -3/2  works.