(x²-3x-4)(x²-3x-18)+24

= (x²-3x)²-22(x²-3x)+72+24

=x²(x²-6x+9)-22x(x-3)+24+72

=x⁴-6x³+9x² -22x²+66x+96

=x⁴-6x³-13x²+66x+96

Verified answer

Answer:

(x² - 3x - 6)×(x² - 3 x - 16)

Step-by-step Explanation:

p(x) = (x+1) (x+3) (x -4) (x -6) + 24

= (x² + 4x +3) (x²- 10x + 24) + 24

= x⁴ - 6 x³ - 13 x² + 66 x +96

Take coefficient of x³ divided by 2, and take it as the coefficient of x, form a square of a quadratic polynomial as below. Value of k is needed to be found. Subtract necessary terms to make it equal to p(x).

p(x) = (x² - 3 x + k)² - [ 2 (k + 11) x² - 6(k+11) x + (k² - 96) ]

Now make the Discriminant of the quadratic polynomial on the right side equal to zero, to make it a perfect square.

D = 9(k+11)² - 2(k+11)(96-k²) = 0.

= (k+11) (2k²+9k -93) = 0.

Take k = -11.

=> [ 2 (k + 11) x² - 6(k+11) x + (k² - 96) ] = 25.

p(x) = (x+1) (x+3) (x -4) (x -6) + 24

= (x² - 3 x - 11)² - 5²

= (x² - 3x -16) (x² - 3x -6)

done.