If a and b are roots of x ² + 5x - 9, then
x ² + 5x - 9 = (x - a) (x - b) = x ² - (a + b) x + ab
so that
a + b = -5
ab = -9
A quadratic with roots 2a + 1 and 2b + 1 would be
(x - (2a + 1)) (x - (2b + 1))
Expanding this gives
x ² - (2a + 1 + 2b + 1) x + (2a + 1) (2b + 1)
The coefficient of the linear term reduces to
-(2a + 1 + 2b + 1) = -2 (a + b + 1) = -2 (-5 + 1) = 8
while the constant term reduces to
(2a + 1) (2b + 1) = 4ab + 2a + 2b + 1 = 4•(-9) + 2•(-5) + 1 = -45
which makes this quadratic equation
x ² + 8x - 45 = 0
