\( 1 \leftarrow \quad \begin{array}{l}\text { Factor } \\ \quad b^{2}+12 b f+32 f^{2}\end{array} \)

To factor the quadratic expression \( b^{2}+12bf+32f^{2} \), we are looking for two numbers that multiply to \( 32f^2 \) and add up to \( 12f \). The numbers \( 4f \) and \( 8f \) work perfectly since \( 4f + 8f = 12f \) and \( 4f \times 8f = 32f^2 \). Thus, we can rewrite the expression as: \[ b^{2} + 4bf + 8bf + 32f^{2} = (b + 4f)(b + 8f) \] So, the fully factored form of the expression is \( (b + 4f)(b + 8f) \). For a quick recap, always look for pairs of numbers that match your needed conditions—this is a key skill in factoring. And remember to check your work by expanding your factors to ensure you return to the original expression!