To find the quadratic equation with roots $-5$ and $1$, and with a leading coefficient of $5$, we start with the general form of a quadratic equation given the roots:

$a(x - r_1)(x - r_2) = 0$

where $r_1$ and $r_2$ are the roots, and $a$ is the leading coefficient.

Here, the roots are $-5$ and $1$, and the leading coefficient $a$ is $5$. Plugging these values in gives:

$5(x - (-5))(x - 1) = 0$

This simplifies to:

$5(x + 5)(x - 1) = 0$

Now, expand the expression:

$5[(x)(x) + (x)(-1) + (5)(x) + (5)(-1)] = 0$

$5(x^2 + 4x - 5) = 0$

Thus, the quadratic equation is:

$5x^2 + 20x - 25 = 0$

Would you like more details or have any questions?

Related Questions:

1. How do you find a quadratic equation when only given the sum and product of the roots?
2. What would the quadratic equation be if the leading coefficient was 1 instead of 5?
3. How does changing the leading coefficient affect the shape of a parabola?
4. Can you derive the quadratic equation from a graph where the roots are known?
5. How do you factor a quadratic equation like $5x^2 + 20x - 25$?

Tip: When given roots of a quadratic equation, remember that the general form is $a(x - r_1)(x - r_2)$, where $a$ can be any non-zero constant, and $r_1$ and $r_2$ are the roots.