To determine the correct factors of the quadratic polynomial [tex]\( x^2 + 3x - 4 \)[/tex], we need to identify two binomials whose product gives us the original quadratic expression. Here is the step-by-step process to factorize it:
1. Identify the quadratic polynomial:
We start with [tex]\( x^2 + 3x - 4 \)[/tex].
2. Find the factors of the constant term [tex]\(-4\)[/tex]:
We need to determine pairs of factors of [tex]\(-4\)[/tex] that will help us break down the middle term ([tex]\(+3x\)[/tex]). These pairs are:
- [tex]\((1, -4)\)[/tex]
- [tex]\((-1, 4)\)[/tex]
- [tex]\((2, -2)\)[/tex]
- [tex]\((-2, 2)\)[/tex]
3. Determine which pair sums to the coefficient of the middle term (3):
The pairs must be evaluated to see which one adds up to the middle coefficient of [tex]\( x \)[/tex], which in this case is [tex]\(+3\)[/tex]:
- [tex]\( 1 + (-4) = -3 \)[/tex]
- [tex]\(-1 + 4 = 3 \)[/tex]
- [tex]\( 2 + (-2) = 0 \)[/tex]
- [tex]\(-2 + 2 = 0 \)[/tex]
Hence, the pair [tex]\((-1, 4)\)[/tex] correctly adds up to [tex]\( 3 \)[/tex].
4. Express the middle term [tex]\( 3x \)[/tex] using the determined pair:
We can write [tex]\( x^2 + 3x - 4 \)[/tex] as [tex]\( x^2 + 4x - x - 4 \)[/tex].
5. Factor by grouping:
Group the terms to facilitate factoring:
[tex]\[ x^2 + 4x - x - 4 \][/tex]
Group terms separately:
[tex]\[ (x^2 + 4x) + (-x - 4) \][/tex]
Now, factor out the greatest common factor from each group:
[tex]\[ x(x + 4) - 1(x + 4) \][/tex]
6. Factor out the common binomial factor:
[tex]\[ (x + 4)(x - 1) \][/tex]
Therefore, the factors of [tex]\( x^2 + 3x - 4 \)[/tex] are [tex]\( (x + 4) \)[/tex] and [tex]\( (x - 1) \)[/tex].
Among the given options, the correct factors are:
[tex]\[ (x + 4) \text{ and } (x - 1). \][/tex]
So, the correct answer is:
[tex]\[ (x + 4) \text{ and } (x - 1). \][/tex]
