Sure! Let's factor the trinomial [tex]\( 5x^2 + 28x + 15 \)[/tex]. We'll go through the following steps to factor this:
1. Identify the coefficients: We have a trinomial of the form [tex]\( ax^2 + bx + c \)[/tex]. Here, [tex]\( a = 5 \)[/tex], [tex]\( b = 28 \)[/tex], and [tex]\( c = 15 \)[/tex].
2. Look for factor pairs of [tex]\( ac = a \cdot c \)[/tex]: We need to find two numbers that multiply to [tex]\( ac \)[/tex] and add up to [tex]\( b \)[/tex]. Here, [tex]\( ac = 5 \cdot 15 = 75 \)[/tex] and we need to find two numbers that multiply to 75 and add up to 28.
Factor pairs of 75:
- 1 and 75 (sum is 76)
- 3 and 25 (sum is 28)
- 5 and 15 (sum is 20)
The factor pair that adds up to 28 is 3 and 25.
3. Rewrite the middle term: Using 3 and 25, rewrite the middle term [tex]\( 28x \)[/tex] as [tex]\( 3x + 25x \)[/tex]:
[tex]\[
5x^2 + 3x + 25x + 15
\][/tex]
4. Factor by grouping: Group the terms to factor by grouping:
[tex]\[
(5x^2 + 3x) + (25x + 15)
\][/tex]
Factor out the greatest common factor (GCF) from each group:
[tex]\[
x(5x + 3) + 5(5x + 3)
\][/tex]
5. Factor out the common binomial: We notice that each group contains the binomial [tex]\( 5x + 3 \)[/tex]:
[tex]\[
(x + 5)(5x + 3)
\][/tex]
Thus, the trinomial [tex]\( 5x^2 + 28x + 15 \)[/tex] factors to:
[tex]\[
(x + 5)(5x + 3)
\][/tex]
This is the factored form of the given trinomial.
