Representing integers efficiently with quadratic polynomials

For every large enough $ T$ given four integers $ a,b,c,d$ with absolute value less than $ T^2$ are there integers $ w_1,x_1,y_1,z_1,w_2,x_2,y_2,z_2$ with absolute value less than $ T$ such that $ $ w_1x_1+y_1z_1=a$ $ $ $ w_2x_1+y_2z_1=b$ $ $ $ w_1x_2+y_1z_2=c$ $ $ $ w_2x_2+y_2z_2=d$ $ holds?

If not for what fraction of $ a,b,c,d$ does it fail?

Typically how many solutions do we get?