Add the following
$4 a^{2} + 5 b^{2} - 8 c^{2}, 7 a^{2} + 3 b^{2}, - b^{2} + c^{2}$

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Solution

To add two or more algebraic expressions, group the like terms of the expression and add each group of like terms separately.
The Sum
$= (4 a^{2} + 5 b^{2} - 8 c^{2}) + (7 a^{2} + 3 b^{2}) + (- b^{2} + c^{2}) = (4 a^{2} + 7 a^{2}) + (5 b^{2} + 3 b^{2} - b^{2}) + (- 8 c^{2} + c^{2})$ (grouping the like terms)
adding the numerical coefficient of like terms
$= (4 + 7) a^{2} + (5 + 3 - 1) b^{2} + (- 8 + 1) c^{2} = 11 a^{2} + (8 - 1) b^{2} + (- 7) c^{2} = 11 a^{2} + 7 b^{2} - 7 c^{2}$

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