To factor the quadratic polynomial [tex]\(2x^2 + 5x + 3\)[/tex] into the product of two binomials, let's go through the step-by-step process.
1. Identify the polynomial: The given polynomial is [tex]\(2x^2 + 5x + 3\)[/tex].
2. Set up the binomials: We need to express this quadratic polynomial in the form [tex]\((ax + b)(cx + d)\)[/tex].
3. Determine [tex]\(a\)[/tex] and [tex]\(c\)[/tex]: Since the coefficient of [tex]\(x^2\)[/tex] is [tex]\(2\)[/tex], we can consider the factors [tex]\(2x\)[/tex] and [tex]\(x\)[/tex] for the terms in the binomials:
[tex]\[
(2x + \_)(x + \_)
\][/tex]
4. Determine [tex]\(b\)[/tex] and [tex]\(d\)[/tex]: We need two numbers that multiply together to give the constant term (3) and add up to the middle coefficient (5).
- The product of the constant terms (b and d) should be 3.
- The sum of the coefficients of x in the binomials should add up to the middle term coefficient (5x).
- Examining the pairs of factors of 3: (1, 3) or (-1, -3). We should consider the positive pair since we need a positive term in the middle.
Let's try:
[tex]\[
(2x + 3)(x + 1)
\][/tex]
5. Verify by expanding:
Expanding [tex]\((2x + 3)(x + 1)\)[/tex]:
[tex]\[
(2x + 3)(x + 1) = 2x \cdot x + 2x \cdot 1 + 3 \cdot x + 3 \cdot 1 = 2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3
\][/tex]
The expanded form matches the original polynomial [tex]\(2x^2 + 5x + 3\)[/tex].
6. Conclusion: The correct factorization is [tex]\((2x + 3)(x + 1)\)[/tex].
So, the correct factors are:
[tex]\(( 2x + 3 )( x + 1 )\)[/tex]
