To calculate: The partial decomposition of \frac{5x+22}{x^{2}+8x+16

Formula used:
Decomposition of ${\displaystyle \frac{f\left(x\right)}{g\left(x\right)}}$ into Partial Fractions:
Consider a rational expression ${\displaystyle \frac{f\left(x\right)}{g\left(x\right)}}$, where f(x) and g(x) are polynomial with real coefficients, ${\displaystyle g\left(x\right)\ne 0}$, and the degree of f(x) is less than degree of g(x).
Step 1: Factor the denominator g (x) completely into linear factors of the form ${\displaystyle {(ax+b)}^m}$ and quadratic factors of the form ${\displaystyle {(a{x}^2+bx+c)}^n}$ that are not further factorable over the integers.
Step 2: Set up the form of decomposition. That is, write the original rational expression ${\displaystyle \frac{f\left(x\right)}{g\left(x\right)}}$ as a sum of simpler fractions using these guidelines. Note that ${\displaystyle A_1,A_2,\dots \dots A_m,B_1,B_2,\dots ..B_m,}$ and ${\displaystyle C_1,C_2,\dots ..C_m,}$ are constants.
Linear Factors of g(x):
For each linear factor of g(x), the partial fraction decomposition must include the sum:
${\displaystyle \frac{A_1}{{(ax+b)}^1}+\frac{A_2}{{(ax+b)}^2}+\dots +\frac{A_m}{{(ax+b)}^m}}$
Quadratic Factors of g(x):
For each quadratic factor of g(x), the partial fraction decomposition must include the sum:
${\displaystyle \frac{B_1}{{(a{x}^2+bx+c)}^1}+\frac{B_2}{{(a{x}^2+bx+c)}^2}+\dots +\frac{B_m}{{(a{x}^2+bx+c)}^m}}$
Step 3: With the form of the partial fraction decomposition set up, multiply both sides of the equation by the LCD to clear fractions.
Step 4: Using the equation from step 3, set up a system of linear equations by equating the constant terms and equating the coefficients of like powers of x.
Step 5: Solve the system of equations from step4 and substitute the solutions to the system into the partial fraction decomposition.
Calculation:
Consider the provided expression:
${\displaystyle \frac{5x+22}{x^2+8x+16}}$
Here, ${\displaystyle f\left(x\right)=5x+22}$ and ${\displaystyle g\left(x\right)=\{x^2+8x+16\}}$.
Factorize g(x):
${\displaystyle g\left(x\right)=\{x^2+8x+16\}}$
${\displaystyle =x(x+4)+4(x+4)}$
${\displaystyle =(x+4)(x+4)}$
${\displaystyle ={(x+4)}^2}$
So, the expression becomes:
${\displaystyle \frac{5x+22}{{(x+4)}^2}}$
Here, in g(x), there is one factor with square power. So, the expression can be decomposed as:
${\displaystyle \frac{5x+22}{{(x+4)}^2}=\frac{A}{(x+4)}+\frac{B}{{(x+4)}^2}}$
Here,