### Instruction: Let $a$ and $b$ be the real roots of
\[x^4 - 4x - 1 = 0.\]Find $ab + a + b.$.### Response: I notice that this is a quartic equation, so it might be hard to find the exact values of $a$ and $b$.
Instead, I will try to use some properties of the roots to find their sum and product, which might help me simplify the expression $ab + a + b$.
One property I remember is that the sum of the roots of a polynomial equation is equal to the opposite of the coefficient of the second-highest degree term, divided by the leading coefficient.
In this case, the second-highest degree term is $x^3$, which has a coefficient of $0$, so the sum of the roots is $0$.
That means that $a + b = 0$, or $b = -a$.
Another property I remember is that the product of the roots of a polynomial equation is equal to the constant term, divided by the leading coefficient.
In this case, the constant term is $-1$, and the leading coefficient is $1$, so the product of the roots is $-1$.
That means that $ab = -1$.
Now I can substitute these values into the expression $ab + a + b$.
I get $ab + a + b = -1 + a - a = -1$.