Question

Given that

(x-p)(2x²+9x+10)=(x²-4)(2x+q)
for all values of x, find the constants p and q.

Answer 1

(x - p)(2x² + 9x + 10) = (x² - 4)(2x + q)
2x³ + 9x² + 10x - 2px² - 9px - 10p = 2x³ + qx² - 8x - 4q
-2x³ -2x³
9x² + 10x - 2px² - 9px - 10p = qx² - 8x - 4q
+8x +8x
9x² + 18x - 2px² - 9px = qx² - 4q

Answer 2

`(x-p)(2x^2+9x+10)=(x^2-4)(2x+q)\\ (x-p)(2x^2+4x+5x+10)=(x-2)(x+2)(2x+q)\\ (x-p)(2x(x+2)+5(x+2))=(x-2)(x+2)(2x+q)\\ (x-p)(2x+5)(x+2)=(x-2)(x+2)(2x+q)\\ (x-p)(x+2)(2x+5)=(x-2)(x+2)(2x+q)\\ \Downarrow\\ p=2\\ q=5`