### 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: We know that if a polynomial has real coefficients, then it must have complex roots in complex conjugate pairs.
That means the complex conjugate of $3+i$, which is $3-i$, must also be a root.
The polynomial must be in the form $a(x-3-i)(x-3+i)$, where $a$ is a nonzero constant.
But we know that the coefficient of $x^2$ is $2$, so $a = 2$.
That means the polynomial is $2(x-3-i)(x-3+i)$.
If we multiply out the polynomial, we get $2(x^2 - 6x + 10)$.