Question

How do you simplify $\left( 4x+7 \right)\left( 3x+2 \right)$ ?

Answer 1

Hint: In this problem, we have to apply the distributive property repeatedly. At last, we get three types of terms, one of ${{x}^{2}}$ , one of $x$ and the third will be an arithmetic term. We then add the $x$ terms together and write the final answer.

Complete step by step solution:
The given expression that we have at our disposal is,
$\left( 4x+7 \right)\left( 3x+2 \right)$
As the above expression involves the multiplications of two expressions within brackets, we use the distributive property. The distributive property states that an expression of the form $a\left( c+d \right)$ can be written as $ac+ad$ . Here, $a=\left( 4x+7 \right),c=3x,d=2$ . Applying this to the above expression, the expression thus becomes,
$\Rightarrow \left( 4x+7 \right)\times 3x+\left( 4x+7 \right)\times 2$
Now, we again apply the distributive property to the first group of the above expression. Then, $a=3x,c=4x,d=7$ . Applying this to the above expression, the expression thus becomes,
$\Rightarrow \left( 4x\times 3x+7\times 3x \right)+\left( 4x+7 \right)\times 2$
Now, we again apply the distributive property to the second group of the above expression. Then, $a=2,c=4x,d=7$ . Applying this to the above expression, the expression thus becomes,
$\Rightarrow \left( 4x\times 3x+7\times 3x \right)+\left( 4x\times 2+7\times 2 \right)$
Multiplying the terms within the brackets, we get,
$\Rightarrow \left( 12{{x}^{2}}+21x \right)+\left( 8x+14 \right)$
Opening up the brackets in the above expression , the above expression thus becomes,
$\Rightarrow 12{{x}^{2}}+21x+8x+14$
Adding the two terms containing $x$ , the above expression thus become,
$\Rightarrow 12{{x}^{2}}+29x+14$
Therefore, we can conclude that the given expression can be simplified to $12{{x}^{2}}+29x+14$ .

Note:
The given problem requires distributive property application numerous times, and that too of unlike terms. Though there are no negative signs involved in this expression, we should be careful for the cases where they are present. At last, we should remember to add or subtract all the like terms, or our answers though being correct, will not be the most simplified one.