Sure, let's find the product step-by-step for the given expression [tex]\((5 - 2x)(3 + x)\)[/tex].
### Step 1: Use the Distributive Property (also known as FOIL method)
To multiply two binomials, we use the distributive property. This method is often called the FOIL method, which stands for First, Outer, Inner, and Last terms.
### The general approach for multiplication:
[tex]\[
(a + b)(c + d) = ac + ad + bc + bd
\][/tex]
For our expression, [tex]\(a = 5\)[/tex], [tex]\(b = -2x\)[/tex], [tex]\(c = 3\)[/tex], and [tex]\(d = x\)[/tex].
### Step 2: Multiply the First terms
[tex]\[
5 \cdot 3 = 15
\][/tex]
### Step 3: Multiply the Outer terms
[tex]\[
5 \cdot x = 5x
\][/tex]
### Step 4: Multiply the Inner terms
[tex]\[
-2x \cdot 3 = -6x
\][/tex]
### Step 5: Multiply the Last terms
[tex]\[
-2x \cdot x = -2x^2
\][/tex]
### Step 6: Combine all the results
[tex]\[
(5-2x)(3+x) = 15 + 5x - 6x - 2x^2
\][/tex]
### Step 7: Simplify the expression
Combine like terms:
[tex]\[
15 + 5x - 6x - 2x^2 = 15 - x - 2x^2
\][/tex]
### Final Answer
The product of [tex]\((5 - 2x)(3 + x)\)[/tex] is:
[tex]\[
-2x^2 - x + 15
\][/tex]
Therefore, [tex]\((5 - 2x)(3 + x) = -2x^2 - x + 15\)[/tex].
