To factor the polynomial [tex]\( x^2 + 9x + 20 \)[/tex], we follow these steps:
1. Identify coefficients:
- The quadratic polynomial is in the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = 20 \)[/tex].
2. Find two numbers that multiply to [tex]\( a \cdot c \)[/tex] and add to [tex]\( b \)[/tex]:
- We need two numbers that multiply to [tex]\( 1 \times 20 = 20 \)[/tex] and add up to [tex]\( 9 \)[/tex].
3. Determine the pair of numbers:
- The pairs of numbers that multiply to 20 are:
- [tex]\( 1 \times 20 \)[/tex]
- [tex]\( 2 \times 10 \)[/tex]
- [tex]\( 4 \times 5 \)[/tex]
- Of these pairs, the one that adds up to [tex]\( 9 \)[/tex] is [tex]\( 4 \)[/tex] and [tex]\( 5 \)[/tex].
4. Write the polynomial in its factored form:
- The factors of [tex]\( x^2 + 9x + 20 \)[/tex] are [tex]\( x + 4 \)[/tex] and [tex]\( x + 5 \)[/tex].
5. Construct the factored form:
- Therefore, the factored form of [tex]\( x^2 + 9x + 20 \)[/tex] is [tex]\( (x + 4)(x + 5) \)[/tex].
So, the factored form of the polynomial [tex]\( x^2 + 9x + 20 \)[/tex] is:
[tex]\[ (x + 4)(x + 5) \][/tex]
And from the given options, the correct answer is:
[tex]\[ (x+4)(x+5) \][/tex]
