How do you multiply $\left( {3x + 4} \right)\left( {2x + 3} \right)$?

Hint: In this question we are asked to multiply the two terms and this can be done by using the FOIL method and the Foil method is a technique used to help remember the steps required to multiply two binomials. Multiply each term in the first binomial with each term in the second binomial using Foil method as shown,
\[\left( {ax + b} \right)\left( {cx + d} \right) = ax \cdot cx + ax \cdot d + b \cdot cx + b \cdot d\].
Now substituting the values and multiplying each term we will get the required result.

Complete step-by-step solution:
Two binomials can be multiplied using Foil method, while multiplying the polynomial, each term of one earlier, while multiplying the polynomial, each term of one polynomial needs to be multiplied by another term of the polynomial. The foil method is just a technique of remembering the series or order of finding the product.
Now given expression is $\left( {3x + 4} \right)\left( {2x + 3} \right)$,
Multiply the first terms of both the binomials that are \[3x\] from $3x + 4$ and \[2x\] from \[2x + 3\]. The product of \[3x\] and \[2x\] i.e.,
\[3x \cdot 2x = 6{x^2}\],
Now then we will multiply the outer terms of both the binomials, the product of outer terms that are \[3x\] from \[3x + 4\] and 3 from \[2x + 3\]i.e.,
\[3x \cdot 3 = 9x\],
Now multiply the inner terms of the binomials. The inner terms here are 4 from \[3x + 4\] and \[2x\] from \[2x + 3\]i.e.,
\[4 \cdot 2x = 8x\],
 At last multiply the last terms in each of the two binomials, the last two terms here are 4 from\[3x + 4\] and 3 from \[2x + 3\], so the product will be i.e.,
\[4 \cdot 3 = 12\].
So this is can represented as,
$ \Rightarrow \left( {3x + 4} \right)\left( {2x + 3} \right) = 3x \cdot 2x + 3x \cdot 3 + 4 \cdot 2x + 4 \cdot 3$,
By simplifying we get,
$ \Rightarrow \left( {3x + 4} \right)\left( {2x + 3} \right) = 6{x^2} + 9x + 8x + 12$,
Now by combining the like terms we get,
$ \Rightarrow \left( {3x + 4} \right)\left( {2x + 3} \right) = 6{x^2} + 17x + 12$

$\therefore $By multiplying the two terms in the given question$\left( {3x + 4} \right)\left( {2x + 3} \right)$ we get, $\left( {3x + 4} \right)\left( {2x + 3} \right) = 6{x^2} + 17x + 12$.

Note: Steps of foil method will be: First multiply the first terms, then the outer terms, then the inner terms and finally the last terms.
The product of two positive will be positive.
The product of two negatives will also be positive.
The product of a positive and negative will always be negative.