To factor the quadratic expression [tex]\(4x^2 + x - 5\)[/tex] by grouping, we need to find two numbers that multiply to [tex]\(4x^2 \times (-5) = -20x^2\)[/tex] and add to the middle coefficient, which is [tex]\(x\)[/tex].
Step 1: Identify the coefficients of the quadratic expression [tex]\(4x^2 + x - 5\)[/tex]:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = 1\)[/tex]
- [tex]\(c = -5\)[/tex]
Step 2: Find two numbers that multiply to [tex]\(a \cdot c = 4 \cdot (-5) = -20\)[/tex], and add to [tex]\(b = 1\)[/tex].
Let's list the factor pairs of [tex]\(-20\)[/tex]:
- [tex]\( (1, -20) \)[/tex] which sums to [tex]\(-19\)[/tex]
- [tex]\((-1, 20)\)[/tex] which sums to [tex]\(19\)[/tex]
- [tex]\( (2, -10) \)[/tex] which sums to [tex]\(-8\)[/tex]
- [tex]\((-2, 10)\)[/tex] which sums to [tex]\(8\)[/tex]
- [tex]\( (4, -5) \)[/tex] which sums to [tex]\(-1\)[/tex]
- [tex]\((-4, 5)\)[/tex] which sums to [tex]\(1\)[/tex]
From these pairs, we see that [tex]\(-4\)[/tex] and [tex]\(5\)[/tex] are the numbers that multiply to [tex]\(-20x^2\)[/tex] and add to [tex]\(1\)[/tex]. So, we rewrite the middle term [tex]\(x\)[/tex] as [tex]\(-4x + 5x\)[/tex].
Step 3: Rewrite the expression [tex]\(4x^2 + x - 5\)[/tex] using these values:
[tex]\[4x^2 + x - 5 = 4x^2 - 4x + 5x - 5\][/tex]
Step 4: Group the terms in pairs:
[tex]\[(4x^2 - 4x) + (5x - 5)\][/tex]
Step 5: Factor out the greatest common factor (GCF) from each pair:
[tex]\[4x(x - 1) + 5(x - 1)\][/tex]
Step 6: Factor out the common binomial factor [tex]\((x - 1)\)[/tex]:
[tex]\[(4x + 5)(x - 1)\][/tex]
Thus, Paul should use [tex]\(-4x\)[/tex] and [tex]\(5x\)[/tex] in the squares [tex]\(M_1\)[/tex] and [tex]\(M_2\)[/tex].
Therefore, the correct option is:
- [tex]\({5x \text{ and } -4x}\)[/tex]
Paul should write [tex]\(5x\)[/tex] in [tex]\(M_1\)[/tex] and [tex]\(-4x\)[/tex] in [tex]\(M_2\)[/tex].
