### Instruction: Find the quadratic polynomial, with real coefficients, which has $3 + i$ as a root, and where the coefficient of $x^2$ is 2.### Response: I know that a quadratic polynomial with real coefficients has the form $ax^2 + bx + c$, where $a, b, c$ are real numbers.
I also know that if $3 + i$ is a root, then its complex conjugate $3 - i$ must be another root, by the complex conjugate root theorem.
So I can write the polynomial as $a(x - 3 - i)(x - 3 + i)$, and then expand it to get $a(x^2 - 6x + 10)$.
Since the coefficient of $x^2$ is 2, I can equate it with $a$ and get $a = 2$.