Find a polynomial with integer coefficients that satisfies the given conditions. \[ P(x)=\square \text { has degree } 2 \text { and zeros } 2+i \text { and } 2-i \] \[ I \]

To find the polynomial with integer coefficients of degree 2 that has zeros \( 2+i \) and \( 2-i \), we can start by using the fact that if \( a \) and \( b \) are the roots, the polynomial can be expressed as: \[ P(x) = (x - a)(x - b) \] Thus, \[ P(x) = (x - (2+i))(x - (2-i)) \] Now we can simplify this expression. The product of the conjugate pairs can be calculated as follows: \[ P(x) = (x - 2 - i)(x - 2 + i) = ((x - 2) - i)((x - 2) + i) \] Using the difference of squares: \[ = (x - 2)^2 - i^2 = (x - 2)^2 - (-1) = (x - 2)^2 + 1 \] Now we expand \((x - 2)^2\): \[ = (x^2 - 4x + 4) + 1 = x^2 - 4x + 5 \] Thus, the polynomial we are looking for is: \[ P(x) = x^2 - 4x + 5 \] This polynomial has integer coefficients and meets the specified conditions.