To factor the quadratic expression [tex]\(2x^2 - 6x - 36\)[/tex] completely, let's follow a systematic approach:
1. Identify the quadratic expression: The expression we need to factor is [tex]\(2x^2 - 6x - 36\)[/tex].
2. Factor out the greatest common factor (GCF) if possible: In this expression, the coefficients of all the terms are divisible by 2. Therefore, we can factor out 2 from the entire expression.
[tex]\[
2x^2 - 6x - 36 = 2(x^2 - 3x - 18)
\][/tex]
3. Focus on factoring the quadratic expression inside the parentheses: We need to factor [tex]\(x^2 - 3x - 18\)[/tex]. We are looking for two binomials [tex]\((x - a)(x + b)\)[/tex] such that when expanded, they give us the original quadratic expression within the parentheses.
4. Find the factors of the constant term (-18) that add up to the coefficient of the middle term (-3):
- The pair of numbers that multiply to -18 and add to -3 are -6 and +3.
5. Write down the factored form using these numbers:
[tex]\[
x^2 - 3x - 18 = (x - 6)(x + 3)
\][/tex]
6. Combine the factored form with the GCF we factored out earlier:
[tex]\[
2(x^2 - 3x - 18) = 2(x - 6)(x + 3)
\][/tex]
Thus, the completely factored form of the quadratic expression [tex]\(2x^2 - 6x - 36\)[/tex] is:
[tex]\[
2(x - 6)(x + 3)
\][/tex]
Therefore, the correct option from the choices provided is:
[tex]\[
2(x - 6)(x + 3)
\][/tex]
