Sure, let's factor the quadratic expression [tex]\( x^2 + x - 56 \)[/tex] step-by-step.
1. Identify the quadratic expression:
[tex]\[ x^2 + x - 56 \][/tex]
2. Look for two numbers that multiply to the constant term (-56) and add up to the coefficient of the linear term (1):
- We need to find two numbers, say [tex]\(m\)[/tex] and [tex]\(n\)[/tex], such that:
[tex]\[ m \cdot n = -56 \][/tex]
and
[tex]\[ m + n = 1 \][/tex]
3. Find the factors that fit the criteria:
- After considering the possible pairs of factors of -56, the suitable pair that fits both conditions is [tex]\((-7)\)[/tex] and [tex]\(8\)[/tex], since:
[tex]\[ (-7) \cdot 8 = -56 \][/tex]
and
[tex]\[ (-7) + 8 = 1 \][/tex]
4. Express the quadratic as a product of two binomials using these factors:
- The quadratic expression can be factored into:
[tex]\[ (x - 7)(x + 8) \][/tex]
Therefore, the blanks are filled as:
[tex]\[
(x - 7)(x + 8)
\][/tex]
So the constants that fill the blanks are [tex]\(-7\)[/tex] and [tex]\(8\)[/tex], and the factored form of [tex]\( x^2 + x - 56 \)[/tex] is:
[tex]\[
(x - 7)(x + 8)
\][/tex]
