Find the coefficient of x2 in the following polynomial x+4x+4x+4.

# Find the coefficient of in the following polynomial $\left(x+4\right)\left(x+4\right)\left(x+4\right).$

1. A
64
2. B
12
3. C
0
4. D
1

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### Solution:

The given polynomial is$\left(x+4\right)\left(x+4\right)\left(x+4\right)$.
We know the formula,${\left(a+b\right)}^{3}={a}^{3}+{b}^{3}+3{a}^{2}b+3a{b}^{2}$.
Simplifying the given expression as,
$\left(x+4\right)\left(x+4\right)\left(x+4\right)={\left(x+4\right)}^{3}$
Applying the formula in the above expression where a = x, b = 4, we get,
${\left(x+4\right)}^{3}={x}^{3}+{4}^{3}+3\left({x}^{2}\right)\left(4\right)+3\left(x\right)\left({4}^{2}\right)$
${\left(x+4\right)}^{3}={x}^{3}+64+12{x}^{2}+48x$
We know that the coefficient of ${x}^{m}$  in $a{x}^{m}+\mathit{bx}+c$ is $a$.
Therefore, coefficient of ${x}^{2}$ in ${x}^{3}+64+12{x}^{2}+48x$ is 12.
Hence, option 2 is  correct.

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