How is this expression solved?

Question

How can we prove the following equality ?

$X(Y+\overline{Y}Z)=X(Y+\overline{Y})(Y+Z)$

I guessed that $\,Y\!\cdot\!Y=Y$.

So they must have done:
$X(Y+\overline{Y} Z)=X(Y\!\cdot\!Y+\overline{Y}Z)$.
But still that doesn’t have the same result.

Answer 1

$X\left(Y+\overline YZ\right)=$

$=X\left(Y\cdot1+\overline YZ\right)=$

$=X\left[Y\!\left(Y\!+\!\overline Y\!+\!Z\right)+\overline YZ\right]=\;\color{blue}{\left(\text{indeed }Y\!+\!\overline Y\!+\!Z\!=\!1\!\right)}$

$=X\left[Y\big(Y+\overline Y\big)+\big(Y+\overline Y\big)Z\right]=$

$=X\left(Y+\overline{Y}\right)\big(Y+Z\big)\,$.