Sure, let's factor the trinomial [tex]\(5x^2 + 34x + 24\)[/tex] step-by-step.
1. Identify the coefficients: We start by identifying the coefficients of the trinomial [tex]\(ax^2 + bx + c\)[/tex]:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = 34\)[/tex]
- [tex]\(c = 24\)[/tex]
2. Find two numbers that multiply to [tex]\(a \cdot c\)[/tex] and add up to [tex]\(b\)[/tex]:
- We need two numbers that multiply to [tex]\(5 \cdot 24 = 120\)[/tex] and add up to [tex]\(34\)[/tex].
3. Determine the two numbers:
- After inspecting possible pairs of factors of 120, we find that [tex]\(4\)[/tex] and [tex]\(30\)[/tex] work because:
[tex]\[
4 \cdot 30 = 120 \quad \text{and} \quad 4 + 30 = 34
\][/tex]
4. Rewrite the middle term using these two numbers:
- Rewrite [tex]\(34x\)[/tex] as [tex]\(4x + 30x\)[/tex]:
[tex]\[
5x^2 + 4x + 30x + 24
\][/tex]
5. Factor by grouping:
- Group the terms:
[tex]\[
(5x^2 + 4x) + (30x + 24)
\][/tex]
- Factor out the greatest common factor (GCF) from each group:
[tex]\[
x(5x + 4) + 6(5x + 4)
\][/tex]
6. Factor out the common binomial:
- Notice that [tex]\((5x + 4)\)[/tex] is a common factor:
[tex]\[
(x + 6)(5x + 4)
\][/tex]
So, the completely factored form of the trinomial [tex]\(5x^2 + 34x + 24\)[/tex] is:
[tex]\[
(x + 6)(5x + 4)
\][/tex]
