To factor the quadratic expression [tex]\( b^2 + b - 20 \)[/tex], follow these steps:
1. Identify the form of the quadratic expression: The expression is in the form of [tex]\( b^2 + b - 20 \)[/tex].
2. Find two numbers that multiply to [tex]\(-20\)[/tex] (the constant term) and add up to [tex]\(1\)[/tex] (the coefficient of the middle term). Let’s denote these numbers as [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
3. We need [tex]\( p \times q = -20 \)[/tex] and [tex]\( p + q = 1 \)[/tex].
4. Let's consider possible pairs of numbers:
- [tex]\( (4, -5) \)[/tex]: [tex]\( 4 \times -5 = -20 \)[/tex] but [tex]\( 4 + (-5) = -1 \)[/tex], which does not satisfy the equation.
- [tex]\( (-4, 5) \)[/tex]: [tex]\( -4 \times 5 = -20 \)[/tex] but [tex]\( -4 + 5 = 1 \)[/tex], which satisfies the equation.
5. Since [tex]\( p = 5 \)[/tex] and [tex]\( q = -4 \)[/tex] satisfy both conditions, we can write the quadratic expression as a product of two binomials:
[tex]\[
(b + 5)(b - 4)
\][/tex]
Thus, the factored form of the expression [tex]\( b^2 + b - 20 \)[/tex] is:
[tex]\[
(b - 4)(b + 5)
\][/tex]
So the slots in the question can be filled as:
[tex]\[
(b + 5)(b - 4)
\][/tex]
