\(\underline{a} \cdot[(\underline{b}+\underline{c}) \times(\underline{a}+\underline{b}+\underline{c})]=\) (a) 0 (b) \([\underline{a} \underline{b} \underline{c}]+[\underline{b} \underline{c} \underline{a}]\) (c) \([\underline{\mathrm{a}} \underline{\mathrm{b}} \underline{\mathrm{c}}]\) (d) none of these.

#tag_title# Step 2: Simplify the cross products #tag_content# Now, we will simplify the cross products in the expression: 1. For the first cross product, we have: \[ (\underline{b}+\underline{c}) \times \underline{a} = \underline{b} \times \underline{a} + \underline{c} \times \underline{a} \] 2. For the second cross product, we have: \[ (\underline{b}+\underline{c}) \times (\underline{b}+\underline{c}) = \underline{b} \times \underline{b} + \underline{b} \times \underline{c} + \underline{c} \times \underline{b} + \underline{c} \times \underline{c} \] Since the cross product of a vector with itself is always 0, the expression becomes: \[ (\underline{b}+\underline{c}) \times (\underline{b}+\underline{c}) = \underline{b} \times \underline{c} + \underline{c} \times \underline{b} \] #tag_title# Step 3: Combine the scalar products #tag_content# Now we will substitute from steps 1 and 2 back into the original expression, and simplify it using the distributive property of the scalar product: \[ \underline{a} \cdot[(\underline{b}+\underline{c}) \times (\underline{a}+\underline{b}+\underline{c})] = \underline{a} \cdot[\underline{b} \times \underline{a} + \underline{c} \times \underline{a}] + \underline{a} \cdot[\underline{b} \times \underline{c} + \underline{c} \times \underline{b}] \] Applying the distributive property, we get: \[ \underline{a} \cdot[(\underline{b}+\underline{c}) \times (\underline{a}+\underline{b}+\underline{c})] = \underline{a} \cdot (\underline{b} \times \underline{a}) + \underline{a} \cdot (\underline{c} \times \underline{a}) + \underline{a} \cdot(\underline{b} \times \underline{c}) + \underline{a} \cdot (\underline{c} \times \underline{b}) \] #tag_title# Step 4: Compare with the given options #tag_content# Comparing this expression with the given options: (a) 0 (b) \([\underline{a}\underline{b}\underline{c}]+[\underline{b}\underline{c}\underline{a}]\) (c) \([\underline{a}\underline{b}\underline{c}]\) (d) None of these It does not match any of the options. So, the answer is (d) None of these.