Suppose that the functions \( s \) and \( t \) are defined for all real numbers \( x \) as follows. \[ \begin{array}{r}s(x)=2 x^{2} \\ t(x)=x+5 \\ \text { Write the expressions for }(t+s)(x) \text { and }(t \cdot s)(x) \text { and evaluate }(t-s)(-2) \\ (t-s)(x)=\square\end{array} \] \( \begin{array}{r}(t+s)(x)=\square\end{array} \)

To find \((t+s)(x)\) and \((t \cdot s)(x)\), let's first compute each step: 1. **Calculate \( (t+s)(x) \)**: \[ (t+s)(x) = t(x) + s(x) = (x + 5) + (2x^2) = 2x^2 + x + 5 \] 2. **Calculate \( (t \cdot s)(x) \)**: \[ (t \cdot s)(x) = t(x) \cdot s(x) = (x + 5) \cdot (2x^2) \] Distributing the terms: \[ = 2x^2(x) + 2x^2(5) = 2x^3 + 10x^2 \] 3. **Now evaluate \((t-s)(-2)\)**: \[ (t-s)(x) = t(x) - s(x) = (x + 5) - (2x^2) = -2x^2 + x + 5 \] To find \((t-s)(-2)\): \[ (t-s)(-2) = -2(-2)^2 + (-2) + 5 = -2(4) - 2 + 5 = -8 - 2 + 5 = -5 \] In summary: \[ (t+s)(x) = 2x^2 + x + 5 \] \[ (t \cdot s)(x) = 2x^3 + 10x^2 \] \((t-s)(x) = -2x^2 + x + 5\) and \((t-s)(-2) = -5\).