5x⁴ -30x² - 135
Factorizing 5x⁴ -30x² - 135, Factoring out 5
5( x⁴ -6x² - 27)..............................(a)
Factoring ( x⁴ -6x² - 27).
x⁴ -6x² - 27 = (x²)² -6(x²) - 27
Let y = x²
Therefore (x²)² -6(x²) - 27 = y² -6y - 27
Therefore (a) is the same as:
5( x⁴ -6x² - 27) = 5(y² -6y - 27)...........(b)
y² -6y - 27 is a quadratic expression
Multiplying first and last terms = y² * -27 = -27y²
We think of two factors that multiply up to -27y², and add up to -6y which is the middle term of the quadratic expression.
The factors are : -9y and +3y
Check: (-9y)*(+3y) = -27y², +3y + (-9y) = 3y -9y = -6y
In the expression y² -6y - 27, we replace the middle term -6y with (3y - 9y)
y² -6y - 27
y² +3y - 9y- 27
y(y+3) - 9(y+3)
(y+3)(y-9)
y² -6y - 27 = (y+3)(y-9) ........................(c)
Recall y = x²
(y+3)(y-9) = (x²+3)(x²-9) ........................(d)
But (x²-9) is difference of two squares = x² - 3² = (x-3)(x+3)
(x²+3)(x²-9)
(x²+3)(x-3)(x+3) ...........................(e)
Therefore from (b)
5(x⁴ -6x² - 27) = 5(y² -6y - 27)
5(y² -6y - 27) = 5(y+3)(y-9) = 5(x²+3)(x²-9) = 5(x²+3)(x-3)(x+3)
So 5x⁴ -30x² - 135 is = 5(x²+3)(x-3)(x+3)
I hope this helps.