Sure, let's multiply and simplify the given expression step-by-step.
We start with the expression:
[tex]\[
(5a + b)(7a + b)
\][/tex]
To expand this expression, we use the distributive property, sometimes known as the FOIL method (First, Outside, Inside, Last):
1. First: Multiply the first terms in each binomial:
[tex]\[
5a \cdot 7a = 35a^2
\][/tex]
2. Outside: Multiply the outer terms in the product:
[tex]\[
5a \cdot b = 5ab
\][/tex]
3. Inside: Multiply the inner terms in the product:
[tex]\[
b \cdot 7a = 7ab
\][/tex]
4. Last: Multiply the last terms in each binomial:
[tex]\[
b \cdot b = b^2
\][/tex]
Now add all these products together:
[tex]\[
35a^2 + 5ab + 7ab + b^2
\][/tex]
Combine the like terms [tex]\(5ab\)[/tex] and [tex]\(7ab\)[/tex] (which are the coefficients of [tex]\(ab\)[/tex]):
[tex]\[
5ab + 7ab = 12ab
\][/tex]
So, the simplified expression is:
[tex]\[
35a^2 + 12ab + b^2
\][/tex]
Thus, the result of multiplying [tex]\((5a + b)(7a + b)\)[/tex] and simplifying is:
[tex]\[
35a^2 + 12ab + b^2
\][/tex]
