Sure! Let's proceed by factoring each of the given polynomials step-by-step.
### 1. [tex]\(2x^2 - 8x\)[/tex]
First, look for the greatest common factor (GCF). Here, the GCF is 2x.
[tex]\[ 2x^2 - 8x = 2x(x - 4) \][/tex]
### 2. [tex]\(-3s^2 + 9s\)[/tex]
Again, identify the greatest common factor. The GCF in this case is -3s.
[tex]\[ -3s^2 + 9s = -3s(s - 3) \][/tex]
### 3. [tex]\(4x + 20x^2\)[/tex]
Find the greatest common factor. Here, the GCF is 4x.
[tex]\[ 4x + 20x^2 = 4x(1 + 5x) \][/tex]
### 4. [tex]\(5t - 10t^2\)[/tex]
Find the greatest common factor, which is 5t.
[tex]\[ 5t - 10t^2 = 5t(1 - 2t) \][/tex]
### 5. [tex]\(s^2 + 8s + 12\)[/tex]
For quadratic polynomials of the form [tex]\(as^2 + bs + c\)[/tex], we look for two numbers that multiply to [tex]\(ac\)[/tex] (which is 12) and add up to [tex]\(b\)[/tex] (which is 8). These numbers are 2 and 6.
The polynomial factorizes as:
[tex]\[ s^2 + 8s + 12 = (s + 2)(s + 6) \][/tex]
To summarize, the factored forms are:
1. [tex]\(2x^2 - 8x = 2x(x - 4)\)[/tex]
2. [tex]\(-3s^2 + 9s = -3s(s - 3)\)[/tex]
3. [tex]\(4x + 20x^2 = 4x(5x + 1)\)[/tex]
4. [tex]\(5t - 10t^2 = -5t(2t - 1)\)[/tex]
5. [tex]\(s^2 + 8s + 12 = (s + 2)(s + 6)\)[/tex]
And this is the detailed, step-by-step solution for factoring each of the given polynomials.
