To form a polynomial with given real zeros, you can create factors from each zero in the form $(x - \text{zero})$, then multiply these factors together. Given that the real zeros are $-2$, $-1$, $4$, and $5$, the corresponding factors are:

$(x + 2), (x + 1), (x - 4), (x - 5)$

Now, multiply these factors to form the polynomial:

$f(x) = (x + 2)(x + 1)(x - 4)(x - 5)$

Step 1: Multiply the first two factors

$(x + 2)(x + 1) = x^2 + 3x + 2$

Step 2: Multiply the last two factors

$(x - 4)(x - 5) = x^2 - 9x + 20$

Step 3: Multiply the two results

$(x^2 + 3x + 2)(x^2 - 9x + 20)$

We will now expand this expression:

$x^2(x^2 - 9x + 20) + 3x(x^2 - 9x + 20) + 2(x^2 - 9x + 20)$

Expanding each term:

1. $x^2(x^2 - 9x + 20) = x^4 - 9x^3 + 20x^2$
2. $3x(x^2 - 9x + 20) = 3x^3 - 27x^2 + 60x$
3. $2(x^2 - 9x + 20) = 2x^2 - 18x + 40$

Combine like terms:

$f(x) = x^4 - 9x^3 + 20x^2 + 3x^3 - 27x^2 + 60x + 2x^2 - 18x + 40$

$f(x) = x^4 - 6x^3 - 5x^2 + 42x + 40$

Thus, the polynomial is:

$f(x) = x^4 - 6x^3 - 5x^2 + 42x + 40$

Would you like any more details or have any questions?

Here are 5 related questions to consider:

1. How do you find the roots of a polynomial if only the equation is given?
2. How can you check if a given number is a root of a polynomial?
3. What is the relationship between the degree of a polynomial and its graph?
4. How can you factor a polynomial when the roots are complex?
5. How does the leading coefficient affect the behavior of a polynomial graph?

Tip: The degree of the polynomial determines the maximum number of real zeros or x-intercepts it can have.