Sure, let's factor the given polynomial [tex]\(2x^2 - 6x - 56\)[/tex] completely.
### Step-by-Step Solution:
1. Write down the polynomial:
[tex]$2x^2 - 6x - 56$[/tex]
2. Factor out the greatest common factor (GCF) from all terms:
Notice that each term in the polynomial has a common factor of 2.
[tex]$2(x^2 - 3x - 28)$[/tex]
3. Factor the quadratic expression inside the parentheses:
Now, we need to factor [tex]\(x^2 - 3x - 28\)[/tex].
To do this, we look for two numbers that multiply to [tex]\(-28\)[/tex] (the constant term) and add up to [tex]\(-3\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
These two numbers are [tex]\(-7\)[/tex] and [tex]\(4\)[/tex], because:
[tex]$-7 \cdot 4 = -28 \quad \text{and} \quad -7 + 4 = -3$[/tex]
4. Rewrite the quadratic expression using these factors:
[tex]$x^2 - 3x - 28 = (x - 7)(x + 4)$[/tex]
5. Combine with the factored out GCF:
Now, we include the GCF we factored out initially:
[tex]$2(x - 7)(x + 4)$[/tex]
Thus, the completely factored form of the polynomial [tex]\(2x^2 - 6x - 56\)[/tex] is:
[tex]$2(x - 7)(x + 4)$[/tex]
Among the given options, the correct one is:
[tex]$\boxed{2(x-7)(x+4)}$[/tex]
