Sure, let's go through the process of factoring each polynomial completely.
### 1. [tex]\( 6x^2 + 9x - 21 \)[/tex]
We start by finding the greatest common factor (GCF) of the terms in the polynomial:
[tex]\[ GCF(6x^2, 9x, -21) = 3 \][/tex]
We factor out the GCF:
[tex]\[ 6x^2 + 9x - 21 = 3(2x^2 + 3x - 7) \][/tex]
So, the factored form is:
[tex]\[ \boxed{3(2x^2 + 3x - 7)} \][/tex]
### 2. [tex]\( 4r^5 - 16y^3 + 20r^2 \)[/tex]
We start by finding the GCF of the terms in the polynomial:
[tex]\[ GCF(4r^5, -16y^3, 20r^2) = 4 \][/tex]
We factor out the GCF:
[tex]\[ 4r^5 - 16y^3 + 20r^2 = 4(r^5 - 4y^3 + 5r^2) \][/tex]
We can also factor out -1 to change the signs for simplification:
[tex]\[ 4(r^5 - 4y^3 + 5r^2) = -4(-r^5 - 5r^2 + 4y^3) \][/tex]
So, the factored form is:
[tex]\[ \boxed{-4(-r^5 - 5r^2 + 4y^3)} \][/tex]
### 3. [tex]\( 10y^3 + 25y^2 - 5y^2 \)[/tex]
First, simplify the polynomial by combining like terms:
[tex]\[ 10y^3 + 25y^2 - 5y^2 = 10y^3 + 20y^2 \][/tex]
We start by finding the GCF of the terms in the polynomial:
[tex]\[ GCF(10y^3, 20y^2) = 10y^2 \][/tex]
We factor out the GCF:
[tex]\[ 10y^3 + 20y^2 = 10y^2(y + 2) \][/tex]
So, the factored form is:
[tex]\[ \boxed{10y^2(y + 2)} \][/tex]
### 4. [tex]\( 7ab^5 + 56abc \)[/tex]
We start by finding the GCF of the terms in the polynomial:
[tex]\[ GCF(7ab^5, 56abc) = 7ab \][/tex]
We factor out the GCF:
[tex]\[ 7ab^5 + 56abc = 7ab(b^4 + 8c) \][/tex]
So, the factored form is:
[tex]\[ \boxed{7ab(b^4 + 8c)} \][/tex]
### 5. [tex]\( 8x^4y^4 - 28x^3y^3 + 4x^2y \)[/tex]
We start by finding the GCF of the terms in the polynomial:
[tex]\[ GCF(8x^4y^4, -28x^3y^3, 4x^2y) = 4x^2y \][/tex]
We factor out the GCF:
[tex]\[ 8x^4y^4 - 28x^3y^3 + 4x^2y = 4x^2y(2x^2y^3 - 7xy^2 + 1) \][/tex]
So, the factored form is:
[tex]\[ \boxed{4x^2y(2x^2y^3 - 7xy^2 + 1)} \][/tex]
This completes the factorization of each polynomial. Each polynomial is factored completely and the answers match what you are looking for.
