Originally Posted by

**bobak** Hello Ashley, A lengthy example is not required you just need to clearly define your question.

Given the root of your quadratic you can write. $\displaystyle x^2 + 4x + 7 = (x -\alpha)(x-\beta)$

Expand and compare coefficients to give.

$\displaystyle \alpha+\beta = -4 \ \ \ (1)$

$\displaystyle \alpha \beta = 7 \ \ \ \ \ \ \ \ (2)$

now you require a quadratic such that $\displaystyle \alpha+2\beta$ and $\displaystyle \beta + 2\alpha$ are roots. So your required quadratic can be written as $\displaystyle (x -( \alpha+2\beta))(x-(\beta + 2\alpha))$

$\displaystyle \Rightarrow x^2 -3(\alpha+\beta)x + \alpha \beta +2 \beta^2 + 4 \alpha \beta + 2 \alpha^2$

$\displaystyle \Rightarrow x^2 -3(\alpha+\beta)x + \alpha \beta +2(\beta + \alpha)^2$

Now appropriately substitute (1) and (2) to get your answer.

Bobak