Instructions: Multiply the following polynomials and simplify your answer.

[tex] \left(4b^2 + b - 7\right)\left(5b^2 + 3b + 6\right) [/tex]

Answer: [tex] \square b^4 + \square b^3 + \square b^2 + \square b + \square [/tex]



Answer :

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]