Sum, difference, and product of two functions Suppose that the functions \( s \) and \( t \) are defined for all real numbers \( x \) as follows. \[ \begin{array}{l}s(x)=3 x^{2} \\ t(x)=x-5\end{array} \] Write the expressions for \( (t \cdot s)(x) \) and \( (t-s)(x) \) and evaluate \( (t+s)(-2) \)

Given the functions \( s(x) = 3x^2 \) and \( t(x) = x - 5 \), we need to find the expressions for \( (t \cdot s)(x) \) and \( (t-s)(x) \) and then evaluate \( (t+s)(-2) \). 1. To find \( (t \cdot s)(x) \), we multiply the functions \( t(x) \) and \( s(x) \): \[ (t \cdot s)(x) = t(x) \cdot s(x) = (x - 5) \cdot 3x^2 \] 2. To find \( (t-s)(x) \), we subtract the function \( s(x) \) from the function \( t(x) \): \[ (t-s)(x) = t(x) - s(x) = (x - 5) - 3x^2 \] 3. To evaluate \( (t+s)(-2) \), we substitute \( x = -2 \) into the expression \( (t+s)(x) \): \[ (t+s)(-2) = (t(-2) + s(-2)) = ((-2) - 5 + 3(-2)^2) \] Now, let's calculate the values. Simplify the expression by following steps: - step0: Solution: \(\left(x-5\right)\times 3x^{2}\) - step1: Apply the distributive property: \(x\times 3x^{2}-5\times 3x^{2}\) - step2: Multiply the terms: \(3x^{3}-15x^{2}\) Calculate or simplify the expression \( (x - 5) - 3x^2 \). Simplify the expression by following steps: - step0: Solution: \(\left(x-5\right)-3x^{2}\) - step1: Remove the parentheses: \(x-5-3x^{2}\) Calculate or simplify the expression \( ((-2) - 5 + 3(-2)^2) \). Calculate the value by following steps: - step0: Calculate: \(\left(-2\right)-5+3\left(-2\right)^{2}\) - step1: Remove the parentheses: \(-2-5+3\left(-2\right)^{2}\) - step2: Multiply the terms: \(-2-5+12\) - step3: Calculate: \(5\) The expressions for \( (t \cdot s)(x) \) and \( (t-s)(x) \) are: - \( (t \cdot s)(x) = 3x^3 - 15x^2 \) - \( (t-s)(x) = x - 5 - 3x^2 \) The value of \( (t+s)(-2) \) is 5.