Which expression is equivalent to \( \left(4 x^{5}+11\right)^{2} ? \) \( 16 x^{5}+121 \) \( 16 x^{10}+121 \) \( 16 x^{10}+88 x^{5}+121 \) \( 16 x^{25}+88 x^{5}+121 \)

To expand the expression \( \left(4 x^{5}+11\right)^{2} \), we can use the binomial theorem, which states that \( (a + b)^{2} = a^{2} + 2ab + b^{2} \). Here, \( a = 4x^{5} \) and \( b = 11 \). Calculating each part: - \( a^{2} = (4x^{5})^{2} = 16x^{10} \) - \( b^{2} = 11^{2} = 121 \) - \( 2ab = 2 \times (4x^{5})(11) = 88x^{5} \) Thus, combining these gives: \[ \left(4 x^{5}+11\right)^{2} = 16 x^{10} + 88 x^{5} + 121 \] So, the correct equivalent expression is \( 16 x^{10}+88 x^{5}+121 \). Now, let's dive into some fun and relevant knowledge: Did you know that the use of mathematical expressions like this dates back to ancient civilizations? The Babylonians had a highly sophisticated understanding of algebra, using geometric methods to solve quadratic equations long before the more formal notation was invented! When working with polynomial expressions, one common mistake is to forget to apply the distributive property correctly. Always double-check your multiplications and additions, especially when handling multiple terms - it’s all too easy to overlook a coefficient or a variable!