To factor the trinomial [tex]\(2c^2 + 11c + 5\)[/tex], we'll use the method of factoring by grouping.
Let's break this process down step-by-step:
1. Identify the trinomial: [tex]\(2c^2 + 11c + 5\)[/tex].
2. Look for two numbers that multiply to the product of the coefficient of [tex]\(c^2\)[/tex] term (which is 2) and the constant term (which is 5), and add to the coefficient of the [tex]\(c\)[/tex] term (which is 11).
- We need two numbers that multiply to [tex]\(2 \times 5 = 10\)[/tex] and add to 11.
3. The numbers that meet these criteria are 10 and 1 because [tex]\(10 \times 1 = 10\)[/tex] and [tex]\(10 + 1 = 11\)[/tex].
4. Rewrite the middle term using these two numbers:
[tex]\[ 2c^2 + 10c + c + 5. \][/tex]
5. Group the terms into pairs:
[tex]\[ (2c^2 + 10c) + (c + 5). \][/tex]
6. Factor out the greatest common factor (GCF) from each pair of terms:
[tex]\[ 2c(c + 5) + 1(c + 5). \][/tex]
7. Factor out the common binomial factor [tex]\((c + 5)\)[/tex]:
[tex]\[ (c + 5)(2c + 1). \][/tex]
So, the factorization of the trinomial is:
[tex]\[ (c + 5)(2c + 1). \][/tex]
Thus, the correct numbers to fill in the blanks are:
[tex]\[
(2c + 1)(c + 5).
\][/tex]
