To factor the quadratic expression [tex]\( 2x^2 + 9x + 4 \)[/tex] into the form [tex]\((ax + b)(cx + d)\)[/tex], follow these detailed steps:
1. Identify the coefficients: We have [tex]\(a = 2\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = 4\)[/tex] from the quadratic expression [tex]\( ax^2 + bx + c \)[/tex].
2. Multiply [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
[tex]\[
a \times c = 2 \times 4 = 8
\][/tex]
3. Find two numbers that multiply to [tex]\(8\)[/tex] and add up to [tex]\(9\)[/tex]:
These numbers are [tex]\(8\)[/tex] and [tex]\(1\)[/tex], as:
[tex]\[
8 \times 1 = 8 \quad \text{and} \quad 8 + 1 = 9
\][/tex]
4. Rewrite the middle term [tex]\(9x\)[/tex] using the two numbers we found:
[tex]\[
2x^2 + 9x + 4 = 2x^2 + 8x + x + 4
\][/tex]
5. Group the terms into pairs and factor each pair:
[tex]\[
(2x^2 + 8x) + (x + 4)
\][/tex]
Factor out the common factors in each group:
[tex]\[
2x(x + 4) + 1(x + 4)
\][/tex]
6. Factor out the common binomial factor [tex]\((x + 4)\)[/tex]:
[tex]\[
(2x + 1)(x + 4)
\][/tex]
Thus, the factored form of the quadratic expression [tex]\(2x^2 + 9x + 4\)[/tex] is:
[tex]\[
(2x + 1)(x + 4)
\][/tex]
For the given specific format [tex]\(([?] x + 1)(x + \square)\)[/tex]:
- In [tex]\(([?] x + 1)\)[/tex], the coefficient in front of [tex]\(x\)[/tex] is [tex]\(2\)[/tex]. So, the first blank adjusts to "2".
- In [tex]\((x + \square)\)[/tex], the constant term is [tex]\(4\)[/tex]. So, the second blank adjusts to "4".
Therefore, the correctly factored form is:
[tex]\[
(2x + 1)(x + 4)
\][/tex]
So, filling in the blanks, we get:
[tex]\[
([2] x + 1)(x + [4])
\][/tex]
