Which values of x and y would make the following expression represent a real number? (4 + 5i)(x + yi)

Answer: The value of x and y are all points which satisfies the equation [tex]4y+5x=0[/tex], i.e.,(4,-5).Explanation:The given expression is,[tex](4+5i)(x+yi)[/tex]Use distributive property to simplify the expression.[tex]4(x+yi)+5i(x+yi)[/tex][tex]4x+4yi+5ix+5yi^2[/tex]We know that [tex]i^2=-1[/tex][tex]4x+4yi+5ix-5y[/tex]Combine likely terms,[tex](4x-5y)+i(4y+5x)[/tex]In x+iy, x is the real part is iy is imaginary part. If the given expression represents a real number it means the imaginary part must be 0.[tex]4y+5x=0[/tex]All the points which satisfies the above equation are the values of x and y for which the given expression is a real number.Fom eg. (4,-5)[tex]4(-5)+5(4)=0[/tex][tex]0=0[/tex]LHS=RHS, it means the point satisfies the equation and the value of x and y are (4,-5).