To factorize the quadratic polynomial [tex]\(2x^2 + 5x + 3\)[/tex], we need to find binomials whose product gives the original quadratic expression. We arrive at the factorization by the following steps:
1. Identify the factors of the quadratic polynomial, matched with the correct coefficients.
2. We know that we should find two binomials such that their product results in [tex]\(2x^2 + 5x + 3\)[/tex].
3. The correct factoring of [tex]\(2x^2 + 5x + 3\)[/tex] is determined to be:
[tex]\[
(2x + 3)(x + 1)
\][/tex]
4. To verify this factorization:
- Multiply [tex]\( (2x + 3) \)[/tex] and [tex]\( (x + 1) \)[/tex].
- [tex]\( (2x + 3)(x + 1) = 2x \cdot x + 2x \cdot 1 + 3 \cdot x + 3 \cdot 1 \)[/tex].
- Simplifying the multiplication:
[tex]\[
= 2x^2 + 2x + 3x + 3
\][/tex]
[tex]\[
= 2x^2 + 5x + 3
\][/tex]
Since this matches the original polynomial [tex]\(2x^2 + 5x + 3\)[/tex], the factors are indeed correct.
Therefore, the correct factorization of the given polynomial is:
[tex]\[
(2x + 3)(x + 1)
\][/tex]
So, the correct choice is:
[tex]\[
(2x + 3)(x + 1)
\][/tex]
