Sure, let's factorize the quadratic expression [tex]\( x^2 + 3x + 2 \)[/tex] in a detailed, step-by-step manner.
### Step-by-Step Solution to Factor [tex]\( x^2 + 3x + 2 \)[/tex]:
1. Identify the quadratic expression we need to factor:
[tex]\[
x^2 + 3x + 2
\][/tex]
2. Recognize the standard form of a quadratic expression, which is [tex]\( ax^2 + bx + c \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex].
3. Set up the problem by looking for two numbers that:
- Multiply to [tex]\( ac \)[/tex] (where [tex]\( a = 1 \)[/tex] and [tex]\( c = 2 \)[/tex]), which is [tex]\( 1 \times 2 = 2 \)[/tex].
- Add up to [tex]\( b \)[/tex], which is [tex]\( 3 \)[/tex].
4. Find the pair of numbers that satisfy these conditions. We need two numbers that multiply to [tex]\( 2 \)[/tex] and add up to [tex]\( 3 \)[/tex]. The numbers are [tex]\( 1 \)[/tex] and [tex]\( 2 \)[/tex].
5. Rewrite the middle term (which is [tex]\( 3x \)[/tex]) using the pair of numbers [tex]\( 1 \)[/tex] and [tex]\( 2 \)[/tex]:
[tex]\[
x^2 + 3x + 2 = x^2 + x + 2x + 2
\][/tex]
6. Factor by grouping:
- Group the terms in pairs:
[tex]\[
(x^2 + x) + (2x + 2)
\][/tex]
- Factor out the common factor from each pair:
[tex]\[
x(x + 1) + 2(x + 1)
\][/tex]
7. Factor out the common binomial factor [tex]\( (x + 1) \)[/tex]:
[tex]\[
(x + 1)(x + 2)
\][/tex]
Thus, the quadratic expression [tex]\( x^2 + 3x + 2 \)[/tex] can be factorized as:
[tex]\[
(x + 1)(x + 2)
\][/tex]
This is the factored form of the given quadratic expression.
