Question

How do you multiply \[\left( 4{{x}^{2}}-9 \right)\left( 8{{x}^{2}}+3 \right)\]?

Answer 1

Hint: We use the process of ‘FOIL’. We multiply the terms according to their positions. There are four multiplications to be done. We complete all four multiplications according to the previously mentioned process.

Complete step-by-step solution:
We have been given multiplication of two quadratic equations. We have to do the breakings of the polynomials in order of FOIL. The word FOIL stands for First-Outside-Inside-Last. It is a technique to distribute the multiplication of polynomials.
There are two terms in each polynomial.
We start by multiplying the first terms of \[\left( 4{{x}^{2}}-9 \right)\] and \[\left( 8{{x}^{2}}+3 \right)\]. The terms are $4{{x}^{2}}$ and $8{{x}^{2}}$.
The multiplication gives a result of $4{{x}^{2}}\times 8{{x}^{2}}=32{{x}^{4}}$.
We now multiply the outside terms of \[\left( 4{{x}^{2}}-9 \right)\] and \[\left( 8{{x}^{2}}+3 \right)\]. The terms are $4{{x}^{2}}$ and 3.
The multiplication gives a result of $4{{x}^{2}}\times 3=12{{x}^{2}}$.
Then we multiply the inside terms of \[\left( 4{{x}^{2}}-9 \right)\] and \[\left( 8{{x}^{2}}+3 \right)\]. The terms are $-9$ and $8{{x}^{2}}$.
The multiplication gives the result of $\left( -9 \right)\times 8{{x}^{2}}=-72{{x}^{2}}$.
We end by multiplying the last terms of \[\left( 4{{x}^{2}}-9 \right)\] and \[\left( 8{{x}^{2}}+3 \right)\]. The terms are $-9$ and 3.
The multiplication gives the result of $\left( -9 \right)\times 3=-27$.
Now we add all the terms and get the final solution as
\[\left( 4{{x}^{2}}-9 \right)\left( 8{{x}^{2}}+3 \right)=32{{x}^{4}}+12{{x}^{2}}-72{{x}^{2}}-27=32{{x}^{4}}-60{{x}^{2}}-27\].
Therefore, multiplied value of \[\left( 4{{x}^{2}}-9 \right)\left( 8{{x}^{2}}+3 \right)\] is \[32{{x}^{4}}-60{{x}^{2}}-27\].

Note: Although we have used the FOIL technique to find the multiplied form of \[\left( 4{{x}^{2}}-9 \right)\left( 8{{x}^{2}}+3 \right)\]. We can also verify the result using the arbitrary value of $x$ where $x=2$.
Putting the value in the equation \[\left( 4{{x}^{2}}-9 \right)\left( 8{{x}^{2}}+3 \right)\], we get $\left( 4{{x}^{2}}-9 \right)\left( 8{{x}^{2}}+3 \right)=\left( 4\times 4-9 \right)\left( 8\times 4+3 \right)=7\times 35=245$.
Putting the value in the equation \[32{{x}^{4}}-60{{x}^{2}}-27\], we get $32{{x}^{4}}-60{{x}^{2}}-27=32\times 16-60\times 4-27=245$.
The equation \[\left( 4{{x}^{2}}-9 \right)\left( 8{{x}^{2}}+3 \right)=32{{x}^{4}}-60{{x}^{2}}-27\] is verified.