To factor the polynomial [tex]\(49 n^2 + 168 n + 144\)[/tex], let's follow the step-by-step process.
1. Identify the terms:
- The polynomial is [tex]\(49 n^2 + 168 n + 144\)[/tex].
2. Recognize the structure of the polynomial:
- This is a quadratic polynomial in the form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 49\)[/tex], [tex]\(b = 168\)[/tex], and [tex]\(c = 144\)[/tex].
3. Determine if the polynomial can be factored as a perfect square:
- A polynomial of the form [tex]\(a^2 + 2ab + b^2\)[/tex] can be factored into [tex]\((a + b)^2\)[/tex].
4. Rewrite the polynomial in a suggestive form:
- Notice that [tex]\(49 = 7^2\)[/tex] and [tex]\(144 = 12^2\)[/tex]. Here, we suspect that the middle term [tex]\(168n\)[/tex] may be twice the product of these terms.
5. Check the middle term:
- Compute [tex]\(2 \cdot 7 \cdot 12 = 168\)[/tex].
Since [tex]\(168 n\)[/tex] is indeed twice the product of 7 and 12, the polynomial can be written as:
[tex]\[
(7n + 12)^2
\][/tex]
Therefore, the factorization of the polynomial [tex]\(49 n^2 + 168 n + 144\)[/tex] is:
[tex]\[
(7n + 12)^2
\][/tex]
