To factor the quadratic expression [tex]\( 4x^2 + 4x - 3 \)[/tex], we aim to write the expression as a product of two binomials of the form [tex]\((ax + b)(cx + d)\)[/tex].
1. Observe the quadratic expression: [tex]\( 4x^2 + 4x - 3 \)[/tex].
2. Choose the form of our binomials: Since the coefficient of [tex]\(x^2\)[/tex] is 4, we might try binomials of the form [tex]\((2x + m)(2x + n)\)[/tex]. Our goal is to find [tex]\(m\)[/tex] and [tex]\(n\)[/tex] such that:
- The product of the first terms gives [tex]\(4x^2\)[/tex].
- The product of the last terms gives [tex]\(-3\)[/tex].
- The middle term (when expanded) gives [tex]\(4x\)[/tex].
3. Set up our binomials and match the middle term:
[tex]\[(2x + m)(2x + n) = 4x^2 + (2n + 2m)x + mn\][/tex]
We need:
- [tex]\(mn = -3\)[/tex]
- [tex]\(2n + 2m = 4\)[/tex]
4. Solve for [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
Given [tex]\(2n + 2m = 4\)[/tex], we can simplify it to:
[tex]\[n + m = 2\][/tex]
5. Find pairs that satisfy [tex]\(mn = -3\)[/tex]:
- Possible pairs [tex]\((m, n)\)[/tex] are:
1. [tex]\(m = -1 \,\, \text{and} \,\, n = 3\)[/tex]
2. [tex]\(m = 3 \,\, \text{and} \,\, n = -1\)[/tex]
6. Check which pair adds up to [tex]\(2\)[/tex]:
- For [tex]\(m = -1\)[/tex] and [tex]\(n = 3\)[/tex]:
[tex]\(m + n = -1 + 3 = 2\)[/tex] which matches.
- For [tex]\(m = 3\)[/tex] and [tex]\(n = -1\)[/tex]:
[tex]\(m + n = 3 - 1 = 2\)[/tex] which also matches.
Both pairs satisfy the criteria, and both binomials are essentially the same since multiplication is commutative:
[tex]\[(2x - 1)(2x + 3)\][/tex]
7. Confirming the correct factorization:
The given expression [tex]\(4x^2 + 4x - 3\)[/tex] when factored is:
[tex]\[
(2x - 1)(2x + 3)
\][/tex]
Therefore, the correct answer to the question is:
c) [tex]\((2x + 3)(2x - 1)\)[/tex]
