To multiply the given polynomials \((4b^2 + b - 7)\) and \((5b^2 + 3b + 6)\), we need to use the distributive property (also known as the FOIL method for binomials) to expand the expression step-by-step.
Given:
[tex]\[
(4b^2 + b - 7)(5b^2 + 3b + 6)
\][/tex]
First, distribute each term in the first polynomial by each term in the second polynomial:
1. Multiply \(4b^2\) by each term in the second polynomial:
[tex]\[
4b^2 \times 5b^2 = 20b^4
\][/tex]
[tex]\[
4b^2 \times 3b = 12b^3
\][/tex]
[tex]\[
4b^2 \times 6 = 24b^2
\][/tex]
2. Multiply \(b\) by each term in the second polynomial:
[tex]\[
b \times 5b^2 = 5b^3
\][/tex]
[tex]\[
b \times 3b = 3b^2
\][/tex]
[tex]\[
b \times 6 = 6b
\][/tex]
3. Multiply \(-7\) by each term in the second polynomial:
[tex]\[
-7 \times 5b^2 = -35b^2
\][/tex]
[tex]\[
-7 \times 3b = -21b
\][/tex]
[tex]\[
-7 \times 6 = -42
\][/tex]
Next, sum all these terms together:
[tex]\[
20b^4 + 12b^3 + 24b^2 + 5b^3 + 3b^2 + 6b - 35b^2 - 21b - 42
\][/tex]
Now, combine like terms:
1. \(20b^4\) (the \(b^4\) term only has one component).
2. Combine \(b^3\) terms:
[tex]\[
12b^3 + 5b^3 = 17b^3
\][/tex]
3. Combine \(b^2\) terms:
[tex]\[
24b^2 + 3b^2 - 35b^2 = -8b^2
\][/tex]
4. Combine \(b\) terms:
[tex]\[
6b - 21b = -15b
\][/tex]
5. The constant term is \(-42\) (no other constants to combine).
Thus, the simplified product of the polynomials is:
[tex]\[
20b^4 + 17b^3 - 8b^2 - 15b - 42
\][/tex]
So, the answer is:
[tex]\[
\boxed{20} \, b^4 \, \boxed{+17} \, b^3 \, \boxed{-8} \, b^2 \, \boxed{-15} \, b \, \boxed{-42}
\][/tex]
