Sure, let's go through the steps to factor the trinomial [tex]\(2c^2 + 11c + 5\)[/tex].
1. Identify the form: We are looking to factor this trinomial into the form [tex]\((2c + X)(c + Y)\)[/tex].
2. Product and Sum: We need to find two numbers that:
- Multiply to give the product of the leading coefficient (2) and the constant term (5). Therefore, [tex]\(2 \times 5 = 10\)[/tex].
- Add up to give the middle coefficient, which is 11.
3. Find the pair: The pair of numbers that satisfy these conditions are 10 and 1:
- [tex]\(10 \times 1 = 10\)[/tex]
- [tex]\(10 + 1 = 11\)[/tex]
4. Factorization: Given these numbers, the factors of the trinomial are:
- [tex]\(2c + 1\)[/tex] (corresponding to 1)
- [tex]\(c + 10 / 2 \rightarrow c + 5\)[/tex]
Therefore, the completely factored form of [tex]\(2c^2 + 11c + 5\)[/tex] is [tex]\((2c + 1)(c + 5)\)[/tex].
So, fill in the blanks as follows:
[tex]\[
(2c + \square)(c + \square)
\][/tex]
[tex]\[
(2c + 1)(c + 5)
\][/tex]
Thus, the correct numbers to fill in the blanks are 1 and 5.
