# 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?