Certainly! Let's solve the quadratic expression [tex]\(5x^2 + 19x - 4\)[/tex] by factoring it step-by-step.
Given expression: [tex]\(5x^2 + 19x - 4\)[/tex]
Step 1: Identify which form of factoring it follows. Here, we should find two binomials that when multiplied together will result in the given quadratic expression.
Step 2: We look for two numbers that multiply to [tex]\(a \cdot c = (5) \cdot (-4) = -20\)[/tex], and add up to [tex]\(b = 19\)[/tex].
The pair of numbers that satisfy these conditions are [tex]\(20\)[/tex] and [tex]\(-1\)[/tex] since:
- [tex]\(20 \cdot (-1) = -20\)[/tex]
- [tex]\(20 + (-1) = 19\)[/tex]
Step 3: We rewrite [tex]\(19x\)[/tex] using the numbers we found:
[tex]\[ 5x^2 + 20x - x - 4 \][/tex]
Step 4: Now, group the terms in pairs and factor each group:
[tex]\[ (5x^2 + 20x) + (-x - 4) \][/tex]
[tex]\[ 5x(x + 4) - 1(x + 4) \][/tex]
Step 5: Factor out the common binomial factor [tex]\((x + 4)\)[/tex]:
[tex]\[ (5x - 1)(x + 4) \][/tex]
Thus, the quadratic expression [tex]\(5x^2 + 19x - 4\)[/tex] factors to:
[tex]\[ (5x - 1)(x + 4) \][/tex]
So, the correct factoring is:
(5x - 1)(x + 4)
