To find the product of the polynomials [tex]\((5d - 12)(-7 + 3d)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials). We'll multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms. Here’s a detailed step-by-step solution organized in a table for clarity:
Step 1: List the individual terms to be multiplied
| Row \ Column | [tex]\(-7\)[/tex] | [tex]\(3d\)[/tex] |
|--------------|---------------|--------------|
| [tex]\(5d\)[/tex] | [tex]\(5d \cdot (-7)\)[/tex] | [tex]\(5d \cdot 3d\)[/tex] |
| [tex]\(-12\)[/tex] | [tex]\(-12 \cdot (-7)\)[/tex] | [tex]\(-12 \cdot 3d\)[/tex] |
Step 2: Perform the multiplication
| Row \ Column | [tex]\(-7\)[/tex] | [tex]\(3d\)[/tex] |
|--------------|---------------------|--------------------|
| [tex]\(5d\)[/tex] | [tex]\(-35d\)[/tex] | [tex]\(15d^2\)[/tex] |
| [tex]\(-12\)[/tex] | [tex]\(84\)[/tex] | [tex]\(-36d\)[/tex] |
Step 3: Combine all the products
[tex]\[
(5d - 12)(-7 + 3d) = \underbrace{-35d}_{\text{term1}} + \underbrace{15d^2}_{\text{term2}} + \underbrace{84}_{\text{term3}} + \underbrace{-36d}_{\text{term4}}
\][/tex]
Step 4: Combine like terms
Notice that we have like terms involving [tex]\(d\)[/tex]:
[tex]\[
-35d - 36d = -71d
\][/tex]
So, the expanded form combining all terms will be:
[tex]\[
15d^2 - 71d + 84
\][/tex]
Final Answer:
[tex]\[
(5d - 12)(-7 + 3d) = 15d^2 - 71d + 84
\][/tex]
