What is expand (e^x+2x)^4 ?

The solution to expand (e^x+2x)^4 is e^{4x}+8e^{3x}x+24e^{2x}x^2+32e^xx^3+16x^4

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expand (e^x+2x)^4

expand

(e^{x} + 2 x)^{4}

e^{4 x} + 8 e^{3 x} x + 24 e^{2 x} x^{2} + 32 e^{x} x^{3} + 16 x^{4}

Solution steps

(e^{x} + 2 x)^{4}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = e^{x}, b = 2 x

= i = 0 \sum 4 (i 4) (e^{x})^{(4 - i)} (2 x)^{i}

Expand summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} (e^{x})^{4} (2 x)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (e^{x})^{3} (2 x)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (e^{x})^{2} (2 x)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (e^{x})^{1} (2 x)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (e^{x})^{0} (2 x)^{4}

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (e^{x})^{4} (2 x)^{0} : e^{4 x}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (e^{x})^{3} (2 x)^{1} : 8 e^{3 x} x

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (e^{x})^{2} (2 x)^{2} : 24 e^{2 x} x^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (e^{x})^{1} (2 x)^{3} : 32 e^{x} x^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (e^{x})^{0} (2 x)^{4} : 16 x^{4}

= e^{4 x} + 8 e^{3 x} x + 24 e^{2 x} x^{2} + 32 e^{x} x^{3} + 16 x^{4}

1 + (\frac{e ^{x} - e ^{- x}}{2})^{2}

\frac{1}{x ( x ^{2} + x + 1 )}

(x^{2} + 2 x + 1)^{6}

\frac{( x ^{4} - 2 x ^{3} + 5 x ^{2} - 3 x + 4 )}{( x - 1 ) ( x + 2 )}

\frac{1}{2} (e^{x^{2}} + e^{- x^{2}})

What is expand (e^x+2x)^4 ?
The solution to expand (e^x+2x)^4 is e^{4x}+8e^{3x}x+24e^{2x}x^2+32e^xx^3+16x^4