Sure, let's work through the step-by-step process of factorizing the given expression [tex]\(15(k+g)^2 - 20(k+g)\)[/tex].
1. Identify the common factor:
The terms in the expression are [tex]\(15(k+g)^2\)[/tex] and [tex]\(-20(k+g)\)[/tex]. Notice that both terms contain a common factor of [tex]\((k+g)\)[/tex].
2. Factor out the common factor:
Factor [tex]\((k+g)\)[/tex] out from each term:
[tex]\[
15(k+g)^2 - 20(k+g) = (k+g) [15(k+g) - 20]
\][/tex]
3. Simplify the coefficient inside the brackets:
Inside the brackets, distribute and simplify the terms inside:
[tex]\[
15(k+g) - 20 = 15k + 15g - 20
\][/tex]
4. Combine the factored expression:
Rewrite the expression with the factored out term:
[tex]\[
15(k+g)^2 - 20(k+g) = (k+g)(15k + 15g - 20)
\][/tex]
5. Re-examine the inner factor for common factors:
Upon examining [tex]\(15k + 15g - 20\)[/tex], the terms [tex]\(15k\)[/tex] and [tex]\(15g\)[/tex] share a factor of 15.
Factor out the common factor of 15:
[tex]\[
15k + 15g - 20 = 15(k + g) - 20
\][/tex]
So:
[tex]\[
(k + g) (15 (k + g) - 20)
\][/tex]
6. General form:
We can generalize the factored form in terms of a single variable for simplicity:
Let [tex]\( u = k + g \)[/tex], so the expression becomes:
[tex]\[
15u^2 - 20u
\][/tex]
7. Factorize in terms of u:
Recognize that [tex]\(15u^2 - 20u\)[/tex] has a common factor of [tex]\(5u\)[/tex]:
[tex]\[
15u^2 - 20u = 5u(3u - 4)
\][/tex]
8. Substitute back [tex]\( u = k + g \)[/tex]:
Replace [tex]\(u\)[/tex] back with [tex]\(k+g\)[/tex]:
[tex]\[
5u(3u - 4) \rightarrow 5(k + g)(3(k + g) - 4)
\][/tex]
Thus, the fully factored form of the original expression [tex]\(15(k+g)^2 - 20(k+g)\)[/tex] is:
[tex]\[
5(k+g) (3(k+g) - 4)
\][/tex]
So, the factored form is:
[tex]\[
5(k + g) (3k + 3g - 4)
\][/tex]
