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2\times 1+2\left(-i\right)+5i\times 1+5\left(-1\right)i^{2}
Multiply complex numbers 2+5i and 1-i like you multiply binomials.
2\times 1+2\left(-i\right)+5i\times 1+5\left(-1\right)\left(-1\right)
By definition, i^{2} is -1.
2-2i+5i+5
Do the multiplications.
2+5+\left(-2+5\right)i
Combine the real and imaginary parts.
7+3i
Do the additions.
Re(2\times 1+2\left(-i\right)+5i\times 1+5\left(-1\right)i^{2})
Multiply complex numbers 2+5i and 1-i like you multiply binomials.
Re(2\times 1+2\left(-i\right)+5i\times 1+5\left(-1\right)\left(-1\right))
By definition, i^{2} is -1.
Re(2-2i+5i+5)
Do the multiplications in 2\times 1+2\left(-i\right)+5i\times 1+5\left(-1\right)\left(-1\right).
Re(2+5+\left(-2+5\right)i)
Combine the real and imaginary parts in 2-2i+5i+5.
Re(7+3i)
Do the additions in 2+5+\left(-2+5\right)i.
7
The real part of 7+3i is 7.