To multiply and simplify the given expression [tex]\( \frac{5x + 15}{7x + 28} \cdot \frac{x + 4}{x + 3} \)[/tex], follow these steps:
1. Factor the Numerators and Denominators:
Let's begin by factoring any common terms in the numerators and denominators.
The numerator [tex]\(5x + 15\)[/tex] can be factored as:
[tex]\[
5x + 15 = 5(x + 3)
\][/tex]
The denominator [tex]\(7x + 28\)[/tex] can be factored as:
[tex]\[
7x + 28 = 7(x + 4)
\][/tex]
2. Rewrite the Expression with Factored Forms:
Substituting the factored forms into the expression:
[tex]\[
\frac{5(x + 3)}{7(x + 4)} \cdot \frac{x + 4}{x + 3}
\][/tex]
3. Cancel Common Factors:
Observe that [tex]\((x + 3)\)[/tex] in the numerator of the first fraction and the denominator of the second fraction are the same. Similarly, [tex]\((x + 4)\)[/tex] in the denominator of the first fraction and the numerator of the second fraction are the same. These terms can be canceled out:
[tex]\[
\frac{5\cancel{(x + 3)}}{7\cancel{(x + 4)}} \cdot \frac{\cancel{(x + 4)}}{\cancel{(x + 3)}}
\][/tex]
After canceling, we are left with:
[tex]\[
\frac{5}{7}
\][/tex]
Therefore, the simplified answer is:
[tex]\[
\boxed{\frac{5}{7}}
\][/tex]
