Expand the quotient by partial fractions. \[ \frac{5 x+5}{x^{2}-4 x+4} \]

To expand the given quotient \(\frac{5x + 5}{x^{2} - 4x + 4}\) by partial fractions, we first need to factor the denominator. The expression \(x^2 - 4x + 4\) can be factored as \((x - 2)^2\). Therefore, we rewrite the quotient as: \[ \frac{5x + 5}{(x - 2)^2} \] Now, we express this fraction in terms of partial fractions. Since the denominator is a repeated linear factor, we set up the partial fraction decomposition as follows: \[ \frac{5x + 5}{(x - 2)^2} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} \] Next, we will multiply both sides by \((x - 2)^2\) to eliminate the denominator: \[ 5x + 5 = A(x - 2) + B \] Now, we expand the right-hand side: \[ 5x + 5 = Ax - 2A + B \] Combining like terms, we get: \[ 5x + 5 = Ax + (B - 2A) \] Now we need to find \(A\) and \(B\) by equating coefficients. For the coefficients of \(x\): \[ A = 5 \] For the constant terms: \[ B - 2A = 5 \] Substituting \(A = 5\) into the second equation gives: \[ B - 2(5) = 5 \implies B - 10 = 5 \implies B = 15 \] Now that we have both constants, we can write the original fraction as: \[ \frac{5}{x - 2} + \frac{15}{(x - 2)^2} \] So the expansion of the quotient by partial fractions is: \[ \frac{5x + 5}{(x - 2)^2} = \frac{5}{x - 2} + \frac{15}{(x - 2)^2} \]