How do you simplify \[(3 + 8x)(9 - 8x + 7{x^2})\] ?

Hint: In this question, we have to simplify the algebraic expression given to us; the algebraic expression is in terms of x. The given algebraic expression is simplified using the distributive property. The distributive property states that the product of a number with the sum of two other numbers is equal to the product of that number with the first number plus the product of that number with the second number and vice versa, that is, $a(b + c) = ab + ac$ .

Complete step-by-step solution:
We have to simplify \[(3 + 8x)(9 - 8x + 7{x^2})\]
We know that when two brackets are in multiplication with each other then we multiply the first term of the first bracket with the whole second bracket and then we multiply the next term of the first bracket with the whole second bracket, that is, $(a + b)(c + d) = a(c + d) + b(c + d)$ , so we get –
\[(3 + 8x)(9 - 8x + 7{x^2}) = 3(9 - 8x + 7{x^2}) + 8x(9 - 8x + 7{x^2})\]
On applying the distributive property, we get –
\[(3 + 8x)(9 - 8x + 7{x^2}) = 27 - 24x + 21{x^2} + 72x - 64{x^2} + 56{x^3}\]
We will place the like terms together –
\[
  (3 + 8x)(9 - 8x + 7{x^2}) = 27 - 24x + 72x + 21{x^2} - 64{x^2} + 56{x^3} \\
   \Rightarrow (3 + 8x)(9 - 8x + 7{x^2}) = 27 + 48x - 43{x^2} + 56{x^3} \\
 \]
Hence the simplified form of \[(3 + 8x)(9 - 8x + 7{x^2})\] is \[27 + 48x - 43{x^2} + 56{x^3}\] .

Note: The given expression is called an algebraic expression because it is a combination of numerical values and alphabets. The alphabets in an algebraic expression represent some unknown quantity, so “x” represents some unknown variable quantity in the given expression. The alphabets and numerical values are linked to each other by some arithmetic operations like addition, subtraction, multiplication and division. We have simplified the given algebraic expression by using the distributive property then we applied the given arithmetic operations to further simplify it. We can also find the value of the unknown variable quantity with the help of an algebraic equation.