To factor the expression [tex]\(4n^2 - 8n - 140\)[/tex], we'll follow a systematic approach. Here's the step-by-step solution:
1. Identify the quadratic expression:
[tex]\[4n^2 - 8n - 140\][/tex]
2. Look for common factors:
First, we notice that all terms in the expression share a common factor of 4. We can factor out a 4 from each term:
[tex]\[4(n^2 - 2n - 35)\][/tex]
3. Factor the quadratic expression inside the parentheses:
We need to factor [tex]\(n^2 - 2n - 35\)[/tex]. To do this, we look for two numbers that multiply to [tex]\(-35\)[/tex] (the constant term) and add up to [tex]\(-2\)[/tex] (the coefficient of the linear term [tex]\(n\)[/tex]).
The two numbers that satisfy these conditions are [tex]\(-7\)[/tex] and [tex]\(5\)[/tex]. Therefore, we can rewrite the middle term [tex]\(-2n\)[/tex] using these numbers:
[tex]\[n^2 - 2n - 35 = n^2 - 7n + 5n - 35\][/tex]
4. Group the terms and factor by grouping:
[tex]\[n^2 - 7n + 5n - 35 = (n^2 - 7n) + (5n - 35)\][/tex]
Now, factor out the common factor from each group:
[tex]\[= n(n - 7) + 5(n - 7)\][/tex]
5. Factor out the common binomial factor:
Notice that [tex]\((n - 7)\)[/tex] is a common factor:
[tex]\[= (n - 7)(n + 5)\][/tex]
6. Combine all factors:
Now substitute back this factored form into the original expression (factored out by 4):
[tex]\[4(n^2 - 2n - 35) = 4(n - 7)(n + 5)\][/tex]
Therefore, the factored form of the expression [tex]\(4n^2 - 8n - 140\)[/tex] is:
[tex]\[
4(n - 7)(n + 5)
\][/tex]
