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-2\times 5-2\times \left(4i\right)+3i\times 5+3\times 4i^{2}
Multiply complex numbers -2+3i and 5+4i like you multiply binomials.
-2\times 5-2\times \left(4i\right)+3i\times 5+3\times 4\left(-1\right)
By definition, i^{2} is -1.
-10-8i+15i-12
Do the multiplications.
-10-12+\left(-8+15\right)i
Combine the real and imaginary parts.
-22+7i
Do the additions.
Re(-2\times 5-2\times \left(4i\right)+3i\times 5+3\times 4i^{2})
Multiply complex numbers -2+3i and 5+4i like you multiply binomials.
Re(-2\times 5-2\times \left(4i\right)+3i\times 5+3\times 4\left(-1\right))
By definition, i^{2} is -1.
Re(-10-8i+15i-12)
Do the multiplications in -2\times 5-2\times \left(4i\right)+3i\times 5+3\times 4\left(-1\right).
Re(-10-12+\left(-8+15\right)i)
Combine the real and imaginary parts in -10-8i+15i-12.
Re(-22+7i)
Do the additions in -10-12+\left(-8+15\right)i.
-22
The real part of -22+7i is -22.