To factor the expression [tex]\(56k^2 + 48k\)[/tex], we will follow these steps:
1. Identify the Greatest Common Factor (GCF):
We first need to identify the greatest common factor of the coefficients of the terms in the expression. The coefficients are 56 and 48.
The prime factors of 56 are [tex]\(2^3 \times 7\)[/tex].
The prime factors of 48 are [tex]\(2^4 \times 3\)[/tex].
The common factor is [tex]\(2^3 = 8\)[/tex].
Additionally, we have a common variable factor, which is [tex]\(k\)[/tex].
Therefore, the GCF of the entire expression is [tex]\(8k\)[/tex].
2. Factor out the GCF:
Once we have determined the GCF, we can factor it out of each term:
[tex]\[
56k^2 + 48k = 8k \left( \frac{56k^2}{8k} + \frac{48k}{8k} \right)
\][/tex]
3. Simplify the expression inside the parentheses:
We perform the division inside the parentheses:
[tex]\[
56k^2 \div 8k = 7k
\][/tex]
[tex]\[
48k \div 8k = 6
\][/tex]
Substituting these values back into the expression, we get:
[tex]\[
56k^2 + 48k = 8k (7k + 6)
\][/tex]
The fully factored form of the given expression is [tex]\(8k(7k + 6)\)[/tex].
Therefore, the correct choice is C. [tex]\(8k(7k + 6)\)[/tex].
