To solve the problem of finding the product of the two polynomials [tex]\( (5 - 2x)(3 + x) \)[/tex], we will use the distributive property, also known as the FOIL method for binomials. FOIL stands for First, Outer, Inner, Last, which refers to the order in which we will multiply the terms in the binomials.
Let's go step-by-step:
### Step 1: Applying the Distributive Property
We need to multiply each term in the first polynomial by each term in the second polynomial.
1. First terms:
[tex]\[
5 \cdot 3 = 15
\][/tex]
2. Outer terms:
[tex]\[
5 \cdot x = 5x
\][/tex]
3. Inner terms:
[tex]\[
-2x \cdot 3 = -6x
\][/tex]
4. Last terms:
[tex]\[
-2x \cdot x = -2x^2
\][/tex]
### Step 2: Combine Like Terms
Now, let's combine all these products:
[tex]\[
5 \cdot 3 + 5 \cdot x + (-2x \cdot 3) + (-2x \cdot x)
\][/tex]
Substituting the values we calculated:
[tex]\[
15 + 5x - 6x - 2x^2
\][/tex]
### Step 3: Simplify the Expression
Combine the like terms (i.e., terms involving [tex]\(x\)[/tex]):
[tex]\[
15 + (5x - 6x) - 2x^2
\][/tex]
[tex]\[
15 - x - 2x^2
\][/tex]
### Step 4: Writing in Standard Form
The standard form of a polynomial is written with terms in descending order of their exponents. Therefore, our final expression will be:
[tex]\[
-2x^2 - x + 15
\][/tex]
### Conclusion
Thus, the product of the polynomials [tex]\( (5 - 2x)(3 + x) \)[/tex] is:
[tex]\[
-2x^2 - x + 15
\][/tex]
