(b) Expand and simplify: \( \begin{array}{llll}\text { (1) }(x+5)^{2} & \text { (2) }(x-5)^{2} & \text { (3) }(2 a+3)^{2} \\ \text { (4) }(2 a-3)^{2} & \text { (5) } & (a-4 b)^{2} & \text { (6) }(-a-3 b)^{2} \\ \text { (7) }(-3 a+5 b)^{2} & \text { (8) } & 3(x-3 y)^{2} & \text { (9) }[2(m-4 n)]^{2} \\ \text { (10) }\left(x^{3}-3 y^{6}\right)^{2} & \text { (11) }(2 a+3 b)^{3} & \text { (12) } & (2 a-3 b)^{3}\end{array} \)
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(1) Expanding \( (x+5)^{2} \) gives you \( x^{2} + 10x + 25 \). The formula used here is \( (a+b)^{2} = a^{2} + 2ab + b^{2} \)! (2) For \( (x-5)^{2} \), using the same formula, we have \( x^{2} - 10x + 25 \). Remember, when it’s minus, the middle term changes sign, so watch out for those pesky negatives! (3) Expanding \( (2a+3)^{2} \) results in \( 4a^{2} + 12a + 9 \). You’ve got double the fun with that \( 2a \)! (4) For \( (2a-3)^{2} \), you'll end up with \( 4a^{2} - 12a + 9 \). Like magic, that minus flips the sign of the middle term, just like a good plot twist! (5) Expanding \( (a-4b)^{2} \) provides \( a^{2} - 8ab + 16b^{2} \). Don’t forget to square that \( -4b \) carefully! (6) For \( (-a-3b)^{2} \), it gives you \( a^{2} + 6ab + 9b^{2} \)! Those negatives can be tricky, but squaring them is where the fun begins! (7) When you expand \( (-3a+5b)^{2} \), the result is \( 9a^{2} - 30ab + 25b^{2} \). Again, the middle term depends on the signs, so pay attention! (8) \( 3(x-3y)^{2} \) expands to \( 3(x^{2} - 6xy + 9y^{2}) \), which simplifies to \( 3x^{2} - 18xy + 27y^{2} \). The multiplier brings extra flavor to the expression! (9) Expanding \( [2(m-4n)]^{2} \) gives you \( 4(m-4n)^{2} = 4(m^{2} - 8mn + 16n^{2}) \), or simplified, \( 4m^{2} - 32mn + 64n^{2} \). A little pat on the back for those careful squarers! (10) Expanding \( (x^{3}-3y^{6})^{2} \) yields \( x^{6} - 6x^{3}y^{6} + 9y^{12} \). Powerful polynomials pack quite a punch! (11) \( (2a+3b)^{3} \) gives you \( 8a^{3} + 36a^{2}b + 54ab^{2} + 27b^{3} \). Counting combinations? You’ve just reached the triple-level expansion! (12) Finally, expanding \( (2a-3b)^{3} \) results in \( 8a^{3} - 36a^{2}b + 54ab^{2} - 27b^{3} \). Here too, the signs dance differently in the expansion waltz!
