Find sum of roots & product of roots

\begin{align*} \text{For the quadratic equation } & ax^2 + bx + c = 0, \\ \\ \text{Sum of roots} & = -{b \over a} \\ \text{Product of roots} & = {c \over a} \end{align*}

Form quadratic equation

With the sum of roots (SOR) and the product of roots (POR),

$$ x^2 - (\text{SOR})x + (\text{POR}) = 0 $$

Example

The roots of the quadratic equation x² - 5x - 10 = 0 are α and β. Find a quadratic equation whose roots are 2α and 2β.

\begin{align} & \underline{\text{For equation } x^2 - 5x - 10 = 0:} \\ \\ & \text{Sum of roots, } \alpha + \beta = -{b \over a} \\ & \phantom{000000000000000/} = -\left({-5 \over 1}\right) \\ & \phantom{000000000000000/} = 5 \\ \\ & \text{Product of roots, } \alpha \beta = {c \over a} \\ & \phantom{0000000000000000} = {-10 \over 1} \\ & \phantom{0000000000000000} = -10 \\ \\ \\ & \underline{\text{For new quadratic equation:}} \\ \\ & \text{Sum of roots} = 2 \alpha + 2\beta \\ & \phantom{000000000(.} = 2(\alpha + \beta) \\ & \phantom{000000000(.} = 2(5) \\ & \phantom{000000000(.} = 10 \\ \\ & \text{Product of roots} = 2\alpha \times 2\beta \\ & \phantom{0000000000000} = 4 \alpha \beta \\ & \phantom{0000000000000} = 4 (-10) \\ & \phantom{0000000000000} = -40 \\ \\ & x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} = 0 \\ & \phantom{00000000000000000(} x^2 - (10)x + (-40) = 0 \\ & \phantom{000000000000000000000(}x^2 - 10x - 40 = 0 \end{align}