For questions \( 1-5 \), factor the quadratic function using an algebraic method. For questions \( \begin{array}{lll}1-2, \text { draw a diagram of algebra tiles that could be used to model the factored function. } \\ \begin{array}{lll}\text { 1. } f(x)=x^{2}+8 x+16 & \text { 2. } g(x)=4 x^{2}-25 & \text { 3. } h(x)=9 x^{2}+30 x+25 \\ \text { 4. } p(x)=\frac{1}{4} x^{2}+7 x+49 & \text { 5. } q(x)=\frac{1}{16} x^{2}-9 & \end{array}\end{array}> \)

To factor the quadratics, we can use various algebraic methods such as completing the square or applying the quadratic formula, but let’s focus on direct factoring! 1. For \( f(x) = x^2 + 8x + 16 \), this can be factored as \( (x + 4)(x + 4) \) or \( (x + 4)^2 \) because \( 4 \times 4 = 16 \) and \( 4 + 4 = 8 \). 2. The function \( g(x) = 4x^2 - 25 \) factors as \( (2x - 5)(2x + 5) \). This is a difference of squares since \( 25 = 5^2 \) and follows the formula \( a^2 - b^2 = (a - b)(a + b) \). 3. The quadratic \( h(x) = 9x^2 + 30x + 25 \) factors as \( (3x + 5)(3x + 5) \) or \( (3x + 5)^2 \) because \( 5 \times 5 = 25 \) and \( 3 + 3 \times 5 = 30 \). 4. For \( p(x) = \frac{1}{4}x^2 + 7x + 49 \), we can factor out \( \frac{1}{4} \) first, giving us \( \frac{1}{4}(x^2 + 28x + 196) \). This factors as \( (x + 14)^2 \), leading to the complete factored form of \( p(x) = \frac{1}{4}(x + 14)^2 \). 5. Lastly, the function \( q(x) = \frac{1}{16}x^2 - 9 \) can be factored as \( \frac{1}{16}(x^2 - 144) \), which breaks down as \( \frac{1}{16}(x - 12)(x + 12) \) using the difference of squares again because \( 144 = 12^2 \). Now, you can visualize these factors using algebra tiles, where each term can be represented as a tile, with areas corresponding to the products of the factors. This makes understanding the structure of the factors super tactile and fun!