Let me use the datas/informations as posted. If they are wrongly posted, then my answer will be wrong also.

y = 4e^x + e^-x -----------(1)

y = A*cosh(x + ln2) --------------(2)

-----------------

cosh(ln(2))

= [e^ln(2) +e^(-ln(2))]/2 ----(i)

Let e^ln(2) = N

Take the ln of both sides,

ln(2)*ln(e) = ln(N)

ln(2) = ln(N)

2 = N

So, e^ln(2) = 2 -----***

Let e^[-ln(2)] = M

1/[e^ln(2)] = M

1/[2] = M

So, e^[-ln(2)] = 1/2 -----***

Substitute those into the cosh(ln(2)),

cosh(ln(2))

= [e^ln(2) +e^(-ln(2))]/2 ----(i)

= [2 +1/2]/2

= 5/4 or 1.25 ---------***

-----------------------------

sinh(ln(2))

= [e^ln(2) -e^(-ln(2))]/2 ----(ii)

= [2 -1/2]/2

= 3/4 or 0.75 ---------***

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y = A*cosh(x + ln2) --------------(2)

y = A*{cosh(x)cosh[ln(2)] +sinh(x)sinh[ln(2)]}

y = A*{[(e^x +e^-x)/2](5/4) +[(e^x -e^-x)/2](3/4)}

y = (A/8){(e^x +e^-x)(5) +(e^x -e^-x)(3)}

y = (A/8){5e^x +5e^-x +3e^x -3e^-x}

y = (A/8){8e^x +2e^-x}

y = (A/4){4e^x +e^-x} ----------(2a)

y from (2a) = y from (1),

(A/4){4e^x +e^-x} = 4e^x +e^-x

Divide both sides by (4e^x +e^-x),

A/4 = 1

A = 1*4 = 4 --------------------------answer.