\( \frac { 5 a ^ { 2 x } - 10 a ^ { x } + 1 } { a ^ { 2 } - 2 a } \)

To simplify the expression \( \frac { 5 a ^ { 2 x } - 10 a ^ { x } + 1 } { a ^ { 2 } - 2 a } \), we can start by factoring both the numerator and the denominator. The denominator can be factored as: \[ a^2 - 2a = a(a - 2) \] Next, we will focus on the numerator \( 5 a^{2x} - 10 a^x + 1 \). This resembles a quadratic in terms of \( a^x \). We can rewrite it as: \[ 5 (a^x)^2 - 10 (a^x) + 1 \] Now, let's set \( y = a^x \). The expression now becomes: \[ 5y^2 - 10y + 1 \] We can factor this quadratic: \[ 5y^2 - 10y + 1 = (5y - 1)(y - 1) \] using the quadratic formula or by inspection. Substituting back \( y = a^x \), we have: \[ (5a^x - 1)(a^x - 1) \] So, the entire expression simplifies to: \[ \frac{(5a^x - 1)(a^x - 1)}{a(a - 2)} \] You can further analyze this expression for specific values of \( a \) and \( x \) if needed.