Where did I go wrong trying to simplify this down to $(x \times y)$ boolean expression?

Question

Simplify:

$[(x \times y') + x']' $

My attempt:

1. $((x \times y') + x')' $
2. $(x \times y')' \times x''$
3. $(x'+ y'') \times x''$
4. $(x' + y'') \times x$
5. $(x' + y) \times x$
6. $(x'+ y)(x' + x)$
7. $x(x' + y) + x'(x' + x)$
8. $xx'+xy+x'x'+x'x$
9. $0 + xy + x + 0$

The answer is supposed to be: $(x \times y)$

N.B. I am still new to this and learning, I am using the following book:

Discrete Mathematics for Computing / Edition 3
by Peter Grossman

Answer 1

You make a mistake going from 5 to 6: Applying Distribution on $(x'+y)x$ results in $x'x + yx$