If \(\alpha\) and \(\beta\) are the roots of the equation \[2x^2+3x+2 = 0,\] find the equation whose roots are \(\alpha+1\) and \(\beta+1\).

If \(y = f(x)\) has roots \(\alpha\) and \(\beta\), then we can find a function with roots \(\alpha+1\) and \(\beta+1\) by translating this \(1\) unit to the right.

Thus \(y = f(x-1)\) will have roots \(\alpha + 1\), \(\beta + 1\).

So the equation we seek is \(2(x-1)^2 +3(x-1) +2 =0\), or \(2x^2-x+1=0\).

It is interesting to note that this works, even though the roots of the original equation are complex!