To determine the factors of the polynomial [tex]\(2x^2 + 5x + 3\)[/tex], we go through the process of factoring it completely.
In this case, we know the polynomial is factorable, and we need to find which pair of binomials, when multiplied together, will expand back to the original polynomial [tex]\(2x^2 + 5x + 3\)[/tex]. The options given are:
1. [tex]\((2x + 3)(x + 1)\)[/tex]
2. [tex]\((2x - 3)(x - 1)\)[/tex]
3. [tex]\((3x + 2)(x + 1)\)[/tex]
4. [tex]\((3x - 2)(x - 1)\)[/tex]
We'll verify each option by multiplying the binomials and checking if they expand to [tex]\(2x^2 + 5x + 3\)[/tex]. We already know the coefficients involved in the correct factorization.
Let's take the first option [tex]\((2x + 3)(x + 1)\)[/tex]:
1. Multiply the first terms: [tex]\(2x \cdot x = 2x^2\)[/tex]
2. Multiply the outer terms: [tex]\(2x \cdot 1 = 2x\)[/tex]
3. Multiply the inner terms: [tex]\(3 \cdot x = 3x\)[/tex]
4. Multiply the last terms: [tex]\(3 \cdot 1 = 3\)[/tex]
Combine the like terms:
[tex]\[2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3\][/tex]
This confirms that the expression [tex]\((2x + 3)(x + 1)\)[/tex] expands to the polynomial [tex]\(2x^2 + 5x + 3\)[/tex].
Thus, the correct factors of the polynomial [tex]\(2x^2 + 5x + 3\)[/tex] are:
[tex]\[
(2x + 3)(x + 1)
\][/tex]
