Question: 3 — More Challenging Quadratic Factoring
LEARNING OBJECTIVE: Factor a quadratic expression in the form $ax^2+bx+c$.
The factored form of $4x^2+4x-15$ is __________.
a.) $(2x - 3)(2x + 5)$
b.) $(4x + 3)(x - 5)$
c.) $(2x + 3)(2x - 5)$
d.) $(4x - 3)(x + 5)$
1. **State the problem:** Factor the quadratic expression $$4x^2 + 4x - 15$$.
2. **Recall the factoring formula:** For a quadratic $$ax^2 + bx + c$$, we look for two binomials $$(mx + n)(px + q)$$ such that:
- $m imes p = a$
- $n imes q = c$
- $m imes q + n imes p = b$
3. **Identify coefficients:** Here, $a=4$, $b=4$, and $c=-15$.
4. **Find factor pairs:**
- Factors of $a=4$ are $(4,1)$ or $(2,2)$.
- Factors of $c=-15$ are $(3,-5)$, $(-3,5)$, $(5,-3)$, or $(-5,3)$.
5. **Test combinations:**
- Try $(2x + 3)(2x - 5)$:
- $2 imes 2 = 4$ (matches $a$)
- $3 imes -5 = -15$ (matches $c$)
- Sum of outer and inner terms: $2 imes -5 + 3 imes 2 = -10 + 6 = -4$ (does not match $b=4$)
- Try $(2x - 3)(2x + 5)$:
- $2 imes 2 = 4$
- $-3 imes 5 = -15$
- Sum of outer and inner terms: $2 imes 5 + (-3) imes 2 = 10 - 6 = 4$ (matches $b$)
6. **Confirm the factorization:**
- Expand $(2x - 3)(2x + 5)$:
$$4x^2 + 10x - 6x - 15 = 4x^2 + 4x - 15$$
7. **Answer:** The factored form is $(2x - 3)(2x + 5)$, which corresponds to option a.)