A quadratic equation with rational coefficients has 2 + √5 as one root. The equation is:

Irrational roots of a rational quadratic occur in conjugate pairs, so the other root is 2 − √5. Their sum is 4 and their product is $2^2-(\surd 5{)}^2=4-5=-1$, giving $x^2-4x-1=0$. (Writing the product as 4 + 5 = 9 gives the last option, and mishandling the sign of the sum gives $x^2+4x-1=0$.) Check: $(2+\surd 5{)}^2-4$(2 + √5) − 1 = 9 + 4√5 − 8 − 4√5 − 1 = 0.