To find the factorization of the trinomial [tex]\(8x^2 + 10x + 3\)[/tex], we will factor it into the product of two binomials of the form [tex]\((ax + b)(cx + d)\)[/tex].
Given the trinomial:
[tex]\[ 8x^2 + 10x + 3 \][/tex]
we need to determine the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] such that:
[tex]\[ (ax + b)(cx + d) = 8x^2 + 10x + 3 \][/tex]
Let's expand the product of two binomials to compare the coefficients:
[tex]\[ (ax + b)(cx + d) = acx^2 + (ad + bc)x + bd \][/tex]
We need:
- [tex]\( ac = 8 \)[/tex] (the coefficient of [tex]\(x^2\)[/tex])
- [tex]\( ad + bc = 10 \)[/tex] (the coefficient of [tex]\(x\)[/tex])
- [tex]\( bd = 3 \)[/tex] (the constant term)
To identify the correct factorization, let's test the provided options:
A. [tex]\((4x + 3)(2x + 1)\)[/tex]:
[tex]\[ (4x + 3)(2x + 1) = 4x \cdot 2x + 4x \cdot 1 + 3 \cdot 2x + 3 \cdot 1 \][/tex]
[tex]\[ = 8x^2 + 4x + 6x + 3 \][/tex]
[tex]\[ = 8x^2 + 10x + 3 \][/tex]
Expanding this product gives us the original trinomial [tex]\(8x^2 + 10x + 3\)[/tex].
Now let's verify if the other options do not match:
B. [tex]\((4x + 3)(2x + 3)\)[/tex]:
[tex]\[ (4x + 3)(2x + 3) = 4x \cdot 2x + 4x \cdot 3 + 3 \cdot 2x + 3 \cdot 3 \][/tex]
[tex]\[ = 8x^2 + 12x + 6x + 9 \][/tex]
[tex]\[ = 8x^2 + 18x + 9 \][/tex]
C. [tex]\((4x + 2)(3x + 1)\)[/tex]:
[tex]\[ (4x + 2)(3x + 1) = 4x \cdot 3x + 4x \cdot 1 + 2 \cdot 3x + 2 \cdot 1 \][/tex]
[tex]\[ = 12x^2 + 4x + 6x + 2 \][/tex]
[tex]\[ = 12x^2 + 10x + 2 \][/tex]
D. [tex]\((4x + 1)(2x + 3)\)[/tex]:
[tex]\[ (4x + 1)(2x + 3) = 4x \cdot 2x + 4x \cdot 3 + 1 \cdot 2x + 1 \cdot 3 \][/tex]
[tex]\[ = 8x^2 + 12x + 2x + 3 \][/tex]
[tex]\[ = 8x^2 + 14x + 3 \][/tex]
Comparing the expanded forms with the original trinomial, we see that the only correct factored form of [tex]\(8x^2 + 10x + 3\)[/tex] is:
A. [tex]\((4x + 3)(2x + 1)\)[/tex]
