If $\alpha , \beta , \gamma \text { and } \delta$ are the roots of the polynomial equation $\left( x ^ { 2 } - 3 x + 4 \right) \left( x ^ { 2 } + 2 x + 5 \right) = 0$, then the quadratic equation whose roots are $a + \beta + \gamma + \delta$ and $\alpha \beta \gamma \delta$ is

To find the required quadratic equation, we first determine the sum and product of all roots alpha, beta, gamma, and delta from the given factored equation. The sum of all four roots is given by the sum of the roots of the first quadratic plus the sum of the roots of the second quadratic, which equals 3 plus negative 2, giving 1. The product of all four roots is found by multiplying the constant terms of the two quadratics, giving 4 multiplied by 5, which equals 20. Therefore, the new quadratic equation with these values as its roots is x squared minus 1x plus 20 equals 0, or x squared minus x plus 20 equals 0.

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