How to construct a quadratic equation based on the roots of other quadratic equations

To determine the quadratic equation based on the roots of other quadratic equations, Consider the following example:

Example:
Develop a quadratic equation whose roots are twice the roots of the quadratic equation x² - 2x -10 = 0!

Answer:
Suppose that the equations of x² - 2x - 10 = 0 are x₁ and x₂.

From the equation obtained:
a = 1
b = -2
c = -10

so:
x₁ + x₂ = -b/a
x₁ + x₂ = -(-2)/(1)
x₁ + x₂ = 2

x₁ . x₂ = c/a
x₁ . x₂ = -10/1
x₁ . x₂ = -10

Suppose that the roots of the new quadratic equation to be searched are α and β whose roots are twice the known root of the equation or α = 2x₁ and β = 2x₂.

α + β = 2x₁ + 2x₂
α + β = 2(x₁ + x₂)
α + β = 2(2)
α + β = 4

α . β = 2x₁ . 2x₂
α . β = 4x₁ . x₂
α . β = 4(-10)
α . β = -40

Then the new quadratic equation which has the roots of α and β is: