The problem requires us to expand the product of two polynomials: $(-r^2 + 3r - 7)(r^2 - 7)$. We need to express the answer as a polynomial in standard form.

1. Problem Description

The problem requires us to expand the product of two polynomials: $(-r^2 + 3r - 7)(r^2 - 7)$. We need to express the answer as a polynomial in standard form.

2. Solution Steps

We will use the distributive property to multiply the two polynomials:

$(-r^2 + 3r - 7)(r^2 - 7) = -r^2(r^2 - 7) + 3r(r^2 - 7) - 7(r^2 - 7)$.

Now, we distribute within each term:

$-r^2(r^2 - 7) = -r^4 + 7r^2$

$3r(r^2 - 7) = 3r^3 - 21r$

$-7(r^2 - 7) = -7r^2 + 49$

Next, we combine these results:

$-r^4 + 7r^2 + 3r^3 - 21r - 7r^2 + 49$

Finally, we combine like terms and write the polynomial in standard form (decreasing order of exponents):

$-r^4 + 3r^3 + (7r^2 - 7r^2) - 21r + 49 = -r^4 + 3r^3 - 21r + 49$

3. Final Answer

$-r^4+3r^3-21r+49$